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Uvod u algebru realnih i kompleksnih brojeva Prof. dr Lazo Roljić Na primjer, desiće vam se na ispitu sljedeći zadatak (ili sličan): „...Ovo je zadatak za „vađenje“ i nosi duple poene!, ali i (u startu) najmanje dvije ocjene manje od maksimalne ocjene ako se ne da pravilan i tačan odgovor. Pretpostavimo da vaš hipotetički program izvodi slijedeće instrukcije: -broj koji je zamislio vaš kolega, u svojoj memoriji pomnožite sa brojem različitih neparnih cifara (5) u aritmetici. - Od tog umnoška, u svojoj memoriji, oduzmite broj različitih parnih cifara (5) u aritmetici (napomena: nula je paran broj) pomnožen brojem cifara u Bulovoj algebri (2). - Dobivenu razliku podijelite u memoriji s brojem neparnih cifara (5) u aritmetici pomnožen s 2. - U vašoj memoriji stoji da je rezultat 30. a) Koji je broj zamislio vaš kolega? b) Prikažite kompletan račun: Ili, ovakav zadatak: Ostrvo Manhattan plaćeno je indijancima 24 $. Da je taj novac bio uložen kod banke uz kamate od 6,5% na godišnjem nivou, akumulirao bi se iznos otprilike onoliki koliko su danas vrijedne sve nekretnine na ostrvu Manhattan, tj. nekoliko hiljada milijardi dolara (recimo 5 hilj. mlrd $). Izračunajte prije koliko godina se ta prodaja desila ako formula za izračunavanje porasta vrijednosti ukamaćivanjem glasi: In = I0 x 1,065br.godina Pri čemu: In = 5.000 milijardi $ I0 = 24 $ Algebra realnih i kompleksnih brojeva Šta je algebra a šta aritmetika??? Aritimetika (termin nastao od grčke riječi aritmos, “broj”) odnosi se na elementarne aspekte teorije brojeva, umijeća mjerenja i numeričkog izračunavanja (tj. proces sabiranja, oduzimanja, množenja, dijeljenja, potenciranja i korijenovanja). Algebra... Algebra se može opisati kao generalizacija (uopštenje) i proširenje aritimetike. Na primjer, činjenica da kada na 2 dodamo 3 daje isti rezultat kao kada na 3 dodamo 2 je aritmetički odnos (2+3=3+2); Formula a+b = b+a, za sve brojeve a,b je algebarski odnos (veza, izraz). Aritmetika postaje algebrom kada se postave opšta pravila koje se odnose na operacije + i * (tzv. Zakon komutacije sabiranja). Natural numbers.. Prvo su postojali prirodni brojevi (natural numbers) ili pozitivni cijeli brojevi: 0, 1, 2, 3, …… (Neki autori isključuju 0). Prirodnim brojevima zovemo pozitivne cijele brojeve . Skup prirodnih brojeva u matematici označavamo velikim slovom N. Skup se često proširuje brojem nula (0) te ga u tom slučaju označavamo sa N0. Algebra of real and complex numbers.. The natural numbers were needed to count objects. (Example: Sumaran’s letters content, 5,000 years ago, e.g. 3,000 years before New Era started.) Zero and negative integers.. Zero and the negative integers: 0, -1, -2, -3, ….. Were introduced in order to make it possible to subtract any positive integer from any other. Rational numbers.. The number system was eventually expanded to include all rational numbers which have the form p/q, in which p and q are integers (natural numbers and their negatives), with q0; 2 3 4 5 6 5 , , , etc. This was made necessary by the advent of division. Racionalni broj lat. (ratio - omjer, razmjer) je broj nastao dijeljenjem dva cijela broja, npr. 1:2, 1:3, 555:333. Može se napisati - u obliku razlomka, a/b, gdje je a brojnik a b nazivnik (cijeli broj različit od 0) ili - u obliku decimalnog broja, npr. 1/2 = 0,5; 1/3 = 0,3333333333... Racionalni broj može imati oblik beskonačnog decimalnog razlomka, ali uvijek periodičnog (ponavlja se isti broj ili ista sekvenca brojeva). Npr: 2/3 = 0, 6666...; 25/36 = 0, 6944444.... Irrational numbers.. The solution to certain problems and equations led to irrational numbers such as 2 , 3, 5 , 3 Iracionalni brojevi (1) Iracionalni brojevi (2) Znači, ovdje nemamo opciju da se decimala ili skup decimala periodično ponavlja, kao što je npr: 0,345345345... ili 2,3333.... Ili 1,25855555..... već imamo niz decimala koje se nikada ne ponavljaju u istoj sekvenci. Realni brojevi.. Skup ili kolekcija svih iracionalnih i racionalnih brojeva. 2 , 3, 5 , 3 2 3 4 5 6 5 , , , etc. ALGEBRA Algebra Algebra is often described as the generalization of arithmetic. The systematic use of variables, letters used to represent numbers, allows us to communicate and solve a wide variety of real-world problems. For this reason, we begin by reviewing real numbers and their operations. Real Numbers A set is a collection of objects, typically grouped within braces { }, where each object is called an element. When studying mathematics, we focus on special sets of numbers. N={1,2,3,4,5,…} Natural Numbers W={0,1,2,3,4,5,…} Integers Z={…,−3,−2,−1,0,1,2,3,…} Whole Numbers The three periods (…) are called an ellipsis (elipse) and indicate that the numbers continue without bound. A subset (podskup), denoted ⊆, is a set consisting of elements that belong to a given set. Notice that the sets of natural and whole numbers are both subsets of the set of integers and we can write: N⊆Z (prir.br. su podskup svih) i W⊆Z (cijel.br. je podskup svih) Racionalni brojevi Rational numbers, denoted Q, are defined as any number of the form ab where a and b are integers and b is nonzero. We can describe this set using set notation: The vertical line | inside the braces reads, “such that” and the symbol ∈ indicates set membership and reads, “is an element of.” The notation above in its entirety reads, “the set of all numbers ab such that a and b are elements of the set of integers and b is not equal to zero.” Racionalni brojevi Na primjer: The set of integers is a subset of the set of rational numbers, Z⊆Q, because every integer can be expressed as a ratio of the integer and 1. In other words, any integer can be written over 1 and can be considered a rational number. Na primjer: IRACIONALNI BROJEVI Irrational numbers are defined as any numbers that cannot be written as a ratio of two integers. Nonterminating decimals that do not repeat are irrational. Na primjer: REALNI BROJEVI Finally, the set of real numbers, denoted R, is defined as the set of all rational numbers combined with the set of all irrational numbers. Therefore, all the numbers defined so far are subsets of the set of real numbers. In summary, REALNI BROJEVI Natural Numbers – The set of counting numbers { 1, 2, 3, 4, 5, …}. Whole Numbers – Natural numbers combined with zero { 0, 1, 2, 3, 4, 5, …}. Integers – Positive and negative whole numbers including zero {…,−5, −4, −3,−2, −1, 0, 1, 2, 3, 4, 5…}. Rational Numbers – Any number of the form a/b where a and b are integers where b is not equal to zero. Irrational Numbers – Numbers that cannot be written as a ratio of two integers. When comparing real numbers, the larger number will always lie to the right of smaller numbers on a number line. It is clear that 15 is greater than 5, but it might not be so clear to see that −5 is greater than −15. Use inequalities to express order relationships between numbers. < > ≤ ≥ "less than" "greater than" "less than or equal to" "greater than or equal to" It is easy to confuse the inequalities with larger negative values. For example, −120 < −10 “Negative 120 is less than negative 10.” Since −120 lies further left on the number line, that number is less than −10. Similarly, zero is greater than any negative number because it lies further right on the number line. 0 > −59 "Zero is greater than negative 59." Absolute Value – The distance between 0 and the real number a on the number line, denoted |a|. Because the absolute value is defined to be a distance, it will always be positive. Važno je da napomenemo da je |0| = 0. Kompleksni brojevi... Kompleksni brojevi Poznato je da je kvadrat svakog realnog broja nenegativan broj. Zbog toga, na primjer, jednostavna jednačina x2 + 1 = 0 nema rješenja u skupu realnih brojeva. Ovo je ekvivalentno sa tvrđenjem da ne postoji realan broj koji bi bio kvadratni korijen realnog broja - 1. Kompleksni brojevi Ovaj nedostatak skupa realnih brojeva možemo otkloniti na taj način što ćemo proširiti skup brojeva sa kojima radimo. Pri tome ćemo voditi računa o tome da se u novom skupu nađu realni brojevi kao njegov podskup i da se sačuvaju važna svojstva osnovnih operacija (sabiranja i množenja) i relacije jednakosti. Kompleksni brojevi Kompleksni brojevi su izrazi oblika gdje su realni brojevi, a neki simbol, za koje su definisane relacija jednakosti (=) i operacije sabiranja (+) i množenja (•) na slijedeći način: Dijeljenje kompleksnih brojeva se svodi na dijeljenje običnim brojem: Algebra of real and complex numbers Complex numbers.. The solution to certain problems and equations led to complex numbers like: 2+3i, 0-4i, 7+0i, where: i 1 Complex numbers.. The complex numbers are constructed from the real numbers together with a number i, the square of which is -1. These numbers can be added and multiplied by well established rules. Complex numbers.. Added, subtract, and multiply next two complex numbers (5-3i) and (1+2i) Results: (6-i), (4-5i), (11+7i) Algebra of real and complex numbers Historically, these events probably did not occur in the order in which they are listed above. In fact, fractions probably preceded negative numbers in most civilizations. Excercises See next slide. VJEŽBE Dati su slijedeći brojevi: 2 3 4 2i 3 9 4 2 3i 3 6 2 3 7 1) Which are integers? 2) Which are real numbers? 3) Which are rational numbers? 4) Which are irrational numbers? 5) In which group are numbers that don’t belong to any of the previous four groups? MATRIX ALGEBRA MATRIX ALGEBRA Week 2 Matrix - basic definitions In this section and the next, we offer formal definitions of matrices and matrix operations, and state certain important properties of matrix algebra. MATRIX -- Basic definitions You frequently see various types of data presented in the form of rectangular array of numbers, such as the box score of a football, basketball, or baseball game; or the vital statistics of movies stars,etc. What is an array? What is an array? An array is rectangular arrangement of quantities (mathematical elements) in rows and columns. It is another name for a matrix. MATRIX -- Basic definitions Consider the following price chart for a length of pipe of given diameters. 2 in. 4 in. 6 in. copper 50¢ 65¢ 75¢ tin 30¢ 40¢ 45¢ 2 in. 4 in. 6 in. copper 50¢ 65¢ 75¢ tin 30¢ 40¢ 45¢ Such rectangular arrays when subject to certain operations are example of matrices.. They are usually designated by capital letters A , B , C, etc. Definition. A matrix A is a rectangular array of entries, denoted by... a11 a12 ... a1n a a 22 ... a 2 n 21 A . . . . a m1 a m 2 ... a mn a11 a12 ... a1n a a ... a 21 22 2n A . . . . a m1 a m 2 ... a mn The entries of matrix aij often are called the elements of the matrix. a11 a12 ... a1n a a ... a 2n A 21 22 . . . . a m1 a m 2 ... a mn Matrix notation allows us to manipulate large rectangular arrays of numbers as single entities. This frequently simplifies the statements of various operations and relationships. Example 1. Let the matrix A 8 A 1 2 2 1 7 Represent the number of gadgets R, S, and T that factories P and Q can produce in a day, that is, matrix A represents the following production capacity. Factory P Factory Q Gadget R 8 per day 2 per day Gadget S 1 per day 1 per day Gadget T 2 per day 7 per day Gadget R Gadget S Gadget T Factory P 8 per day 1 per day 2 per day Factory Q 2 per day 1 per day 7 per day Later, using the entity “matrix A”, we will extend this example to accomplish some very useful results. 8 A 1 2 2 1 7 Scalars? Scalars... Scalars! In this course we shall assume, unless it is stated otherwise, that the entries of a matrix are scalars ; (no matrices, no vectors) We can interpret the term scalar as a complex number, a real number or a function thereof. Under certain conditions, the entries of a matrix may also be certain other matrices. Matrix is not a scalar !!! Matrix does not represent a number of any type!! A matrix does not have a numerical value; it is merely an array. Examples of matrices.. The following are examples of matrices: The entries are integers, functions of x, complex numbers, real numbers, and matrices. 2 1 2 4 0 1 2 3 5 2 3 2i i 3 i 1 x2 2 2 x x 2 2 4 6 1 0 1 3 0 1 4 2 4 Real matrix.. If all the entries of a matrix are real numbers, it is called real matrix. 2 1 2 4 0 1 2 3 5 2 3 7 5 A 7 0 2 3 7 Zero matrix.. If all of the entries of a matrix are zero, the matrix is called the zero matrix or null matrix and is denoted by 0. Bold print will be used to distinguish the zero matrix from zero scalar. 0 0 0 0 0 0 0 The horizontal lines of the array are called rows. The vertical lines are columns. What is a matrix? So, we can say, A matrix is a rectangular array of numbers (or scalars) arranged in rows and columns. Matrix Each entry is designated in general as aij , where i represents the row number, and j is the column number; thus a31 is the entry in the third row and first column. The double subscript can be called the address of the entry. 2 1 2 A 4 0 1 The dimensions of the array (number of rows stated first) determine the order or shape of the matrix, designated “m by n”. Matrix A in this example has a total 2 rows and 3 columns of entries. We say that the order (shape) of this matrix is 2 by 3. The order of matrix in next example is ? 2 3 5 2 3 The order of this matrix is 3 by 1. 2 3 5 2 3 Such a matrix with of a single column is called a column matrix. When a matrix consists of a single row (is of order 1 by n), it is called a row matrix. 6 3 3 314 . How many columns does this row matrix have ? 6 3 3 314 . Examples.. Example of row-matrix.. F f1 f 2 f 3 Example of matrix g11 g12 g13 G g g g 21 22 23 What is the address of entry g23 ? (2,3) Example of column matrix x1 X x2 x 3 What is the address of entry x2 ? (2,1) Can one subtract or add column and row matrices? Why not? Because they are not conformable for subtraction and addition. Transposition of matrices.. Definition.. The transpose AT of a m by n matrix A=[aij] is the m by n matrix Q=[qij] obtained by interchanging the rows and columns of A, thus AT=Q=[qij] where qij=aji (i=1,2,…,n; j=1,…,m) For example.. then if 1 1 3 A 3 7 4 2 5 1 1 T A 3 2 3 7 4 5 and.. if 5 1 X 0 4 then X 5 1 0 4 T Several important properties of transposes.. 1) The transpose of transpose is the original matrix (AT)T=A 2) The transpose of a sum is the sum of transposes (A+B)T=AT+BT 3) The transpose of a scalar times a matrix is the same scalar times the transpose of that matrix (kA)T=kAT 4) The transpose of a product is the product of the transposes in reverse order: (AB)T = BTAT 5) If A and B are two column vectors (ie, row matrix or column matrix) each of row degree n. Then ATB is a scalar;hence (ATB)T=BT(AT)T=BTA , (ATB)T=BTA called inner product, dot product or scalar product. Square matrix.. When the dimensions of a matrix are equal, it is called a square matrix. 3 0 4 7 2 6 0 0 The main diagonal of a square matrix consists of the entries a11,a22,a33,…,ann 3 0 4 7 2 6 0 0 Does a non-square matrix have a main diagonal? No, it doesn’t!!! Diagonal matrix.. A diagonal matrix is a square matrix whose off-diagonal elements, aij for ij, are equal to zero. Example.. 4 A 0 0 0 7 0 0 0 3 A Unit Matrix, or Identity matrix, or 1 Matrix Identity Matrix A unit matrix, or identity matrix, or 1 matrix, is a diagonal matrix whose diagonal elements are 1. A unit matrix of order n is denoted by In or simply I. For n = 3, we have 1 0 0 I 3 0 1 0 0 0 1 Identity Matrix Can a non-square matrix be a unit matrix? No, it can’t, because non-square matrix has no main diagonal. Only square matrices have an identity matrix! Identity Matrix Hence, the unit matrix is always a square matrix. This 3 by 3 real matrix is said to be of order 3. In general an n by n matrix is said to be of order n. 3 0 4 7 2 6 0 0 This is matrix of order 3. Matrices are denoted in several different ways by different authors. Some use parentheses, some use double vertical lines. In this course we will use one of the three following ways: a11 a12 a a 21 22 A . . a m1 a m 2 ... a1n ... a 2 n aij ... . ... a mn ( m,n ) Notation 9 6 23 A 15 2 0 12 3 / 4 2.7 A 655 . 917 . 44 . 012 . 73 239 A Matrix does not represent a number of any type!!! A Matrix is not a scalar!!! It does not have a value!!! Exercises.. Exercises.. Is the number of rows of a given matrix always the same as the number of columns? No, it isn’t. Exercises.. Is the number of rows of a given matrix always greater than the number of columns? No, it isn’t. Is this a matrix ? Why ? 2 1 0 -2 3 No, it isn’t, because is not a rectangular array. Exercises What is the entry in the third row and second column of matrix A? What is the address of the entry 6? 1 A 6 1 9 2 3 Exercises What is a12 and what is a21 in next matrix? a ij ( 2 ,2 ) 2 2 0 1 2 3 1 4 9 8 7 5 Matrix Equalities and Inequalities Definition: Two matrices A and B are said to be equal when they are of the same order (have the same shape) and all their corresponding entries are equal; that is, aij = bij for all i and j. Matrix Equalities and Inequalities Example: A=B 1 A 6 1 9 2 3 1 B 6 1 9 2 3 What is the condition for A=B in this example? A=B only when x=3. x A 4 1 2 7 3 3 B 4 1 2 7 3 Why does AB in this next example? x A 4 1 2 7 3 x B 4 1 2 7 3 5 1 9 AB because A and B do not have the same order (do not have the same shape). Matrix Inequalities Definition: A real matrix A is said to be “greater than” (>) real matrix B of the same order when each of the entries of A is “greater than” each of the corresponding entries of B. Matrix Inequalities.. Matrices behave differently from real numbers with respect to inequalities. For two real numbers a and b: If ab and ab, then a=b. That is not the case for real matrices of the same order. Matrix nonnegativity.. A matrix is said to be nonnegative if each entry (elements) is said to be nonnegative if each entry (elements) is nonnegative. Thus we write X 0 to indicate that each component xij of X is a nonnegative real number Exercises.. Matrix Equalities and Inequalities Find, if possible, all values for each unknown that will make each of the following true: 1 x 0 4 1 4 y 2 x 1 2 y 3 0 3 Calculate if possible A+B+C=? 1 A 2 4 8 3 2 C 2 2 2 5 B 0 1 Result: 6 7 A BC 0 5 Find, if possible, all values for unknown x that will make of the following true: 2 x 1 0 1 3 1 6 2 5 0 3 Matrix addition and subtraction.. Matrix Addition Matrix addition and subtraction can be performed only when the two matrices to be added are of the same order. We say then that they are conformable for addition or conformable for subtraction. Definition: Given matrices A=[aij](m,n) and B=[bij](m,n) Matrix addition is defined as A+B=[aij+ bij] (m,n) In other words, if two matrices are of the same order they may be added by adding corresponding entries. The same rules are valid for subtracting the matrices. Matrix subtraction is defined as A-B=[aij- bij] (m,n) In other words, if two matrices are of the same order they may be subtracted by subtracting corresponding entries. Calculate if possible A+B=? A 0 5 8 3 B 6 4 Not the same order, hence impossible. Matrix Addition and Subtraction If A is a 2 by 3 matrix and B is a 3 by 2 matrix, are A and B conformable for addition? Are A and B conformable for subtraction? Why? 1 2 A 3 4 Using 2 and B 1 1 3 Prove rule 2) for the transpose of the sum of two matrices : (A+B)T=AT+BT Prove rule 2) for the transpose validity for the case (A-B)T =AT-BT Example: Matrix Addition A manufacturer produces a certain metal. The costs of purchasing and transporting specific amounts of necessary raw materials (ores) from two different locations are given by the following matrices: Example: Matrix Addition p. c. t . c. 16 20 R A 10 16 S 4 9 T p. c. t . c. 12 10 R B 14 14 S 12 10 T p.c. means purchasing costs; t.c.means transport costs R, S and T are ores Find the matrix representing the total p.c. and t.c. of each type of ore! Find the matrix representing the total p.c. and t.c. of each type of ore! p. c. 28 A B 24 21 t . c. 30 R 30 S 14 T MSCI2400 Analytical Methods in Management Dr LAZO ROLJIC Fulbright Fellow DuPree College of Management Faculty of Economics University of Banja Luka Bosnia and Herzegovina Matrix algebra... Week 2-2 Multiplying a matrix by a scalar... Definition: Given matrix A=[aij](m,n) and scalar k, then kA = [k aij](m,n) In other words, a matrix may be multiplied by a scalar by multiplying every entry of the matrix by the scalar. Examples: 3 2 4 A 1 0 2 and k=2 2 3 2 2 2 4 6 4 8 kA 2 1 2 0 2 ( 2) 2 0 4 5 2 6 B 1 0 1 and k=-1 5 kB ( 1) B B 1 2 0 6 1 Is there any restriction on the multiplication of a matrix by scalar? No. The product of a scalar and matrix is another matrix. We are now in position to handle matrix subtraction similarly to the way in which we perform scalar subtraction. Remember that the rule in real number algebra is: a-b = a + (-b) or 6-4=6+(-4) Likewise, for matrices, A-B=A+(-B). Matrix addition is a) Associative: b) Distributive: c) Commutative: (A+B)+C=A+(B+C) k(A+B)=kA+kB (c+k)A=cA+kA A+B=B+A A+0=A Multiplication of Matrices.. The product of two matrices.. We now define a second kind of product the product of two matrices. Before giving a general definition, however, we consider the method of multiplying a row matrix (row-vector) by a column matrix (column vector). (Here the row matrix precedes the column matrix and the number of columns in A equal number of rows in B)! Definition Let A be a 1 by p matrix and B be a p by 1 matrix. The product C=AB is a 1 by 1 matrix given by b11 b 21 . a11 a12 ... a1 p a11b11 a12b21 ...a1 pbp1 . . bp1 For example, 0 2 1 3 4 (2 0) (1 4) (3 2) 10 !!!! 2 Product of row matrix and column matrix is 1 by 1 matrix, that is a SCALAR! Matrix multiplication In order to present some logic to the definition of the product of two matrices, we find it convenient first to define the product of two vectors. The special matrices form consisting of either a single row or a single column are referred to as vectors. Matrix multiplication A vector, whether it be a row vector or a column vector, is a special case of the matrix; in the first case it has one row and in the second case it has one column. In either case it can be considered to be a point in n-dimensional space, where the components are the coordinates. 1 3 The examples of two vectors: 3.5 3 2.5 2 1.5 1 0.5 0 and 2 2 (1,3) (2,2) Series1 0 1 2 3 The examples of two vectors 3.5 3 2.5 2 1.5 1 0.5 0 1 3 and 2 2 (1,3) (2,2) Series1 0 1 2 3 Therefore the row vector [a1 a2 a3] or the column vector a1 a 2 a3 can be considered to be a point in threedimensional space. There is no geometrical distinction between these two vectors. The decision to write a vector as a row or a column vector is therefore a matter of convenience. Matrix multiplication The operation of addition or subtraction of two vectors or multiplication by a scalar (as we will se later) follows the rules for matrices. Specific for vectors are inner product and vector length or norm of vector. Matrix multiplication Notice that the number of columns of A must equal the number of rows of B. (Conformable for multiplication or susceptible to multiplication) One use of this operation may be forecast, if we observe that a linear equation may be expressed using a product of two matrices, that is, x 3 3 y 2 Means [2x+3y]=[3] and therefore 2x + 3y = 3 Now in order to express a system of equations such as 2x+3y = 3, x + 4y = 1, using matrix notation, we will need the following definition. Here we are not restricted to a row matrix times a column matrix, although we will see that the multiplication simply requires a succession of the manipulation previously described. Definition.. Let A be an m by p matrix and B be a p by n matrix. The product C=AB is an m by n matrix where each entry cij of C is obtained by multiplying corresponding entries of the ith row of A by those of the jth column of B and the adding the results. Matrix multiplication The operation defined above can be illustrated in general by the following diagrams. a11 a21 am1 a12 a22 . . am 2 ... a1 p b11 b12 ... a2 p b21 b22 . . . . ... amp bp1 bp 2 ... b1n c11 c12 b ... 2 n c21 c22 . . . . ... bpn cm1 cm2 ... c1n ... c2 n . . ... cmn Matrix multiplication Where c11=a11b11+a12b21+…+a1pbp1 or, generally speaking... Matrix multiplication a 11 . ai1 . am1 ... . ... . ... a1 p b11 . aip . amp bp1 ... bij . . . . . . . . . ... bpj ... b1n c11 ... c1n . . . . . . . . cij . . . . . . ... bpn cm1 ... cmn Matrix multiplication Where cij = ai1b1j + ai2b2j +…+aipbpj One very handy rule.. If you multiply matrix A of dimension (m,n) and matrix B of dimension (n,k), then result matrix will be matrix C of dimension (m,k). You can see that (m,k) are the outer numbers of matrices A and B dimensions. Example: 1 2 A 1 0 3 0 1 B 0 1 1 1 2 3 0 1 AB 1 0 0 1 1 (1 3 2 0) (1 0 2 1) (1 1 2 1) 3 2 3 AB (1 3 0 0) (1 0 0 1) (1 1 0 1) 3 0 1 Find result BA, if possible ?! Why is this impossible? This is because the number of columns of the left matrix does not equal the number of rows of the right matrix. In order for multiplication of matrices to be performed, the number of columns of the left matrix must equal the number of rows of the right matrix. We than say that the left matrix is conformable or susceptible for multiplication to the right matrix. Matrix multiplication Prior to our definition of matrix multiplication we implied that it could be used to express the system of equations 2x + 3y =3 x + 4y =1 Consider the following matrix equation. 2 3 x 3 1 4 y 1 2x + 3y =3 x + 4y =1 2 1 3 x 3 4 y 1 Perform the indicated matrix multiplication on the left side;there results 2 x 3 y 3 x 4 y 1 By the definition of the equality of matrices we obtain our original system. Example 2 AB 3 1 0 2 1 0 4 2 1 B A 1 3 3 2 4 (2 0 11) (2 4 1 3) 1 11 3 (3 0 2 1) (3 4 2 3) 2 18 To form A, B was premultiplied by A;To form BA, B was postmultiplied by A. 0 BA 1 4 2 3 3 1 (0 2 4 3) (0 1 4 2) 12 2 (1 2 3 3) (11 3 2) 11 Note: BAAB. This is quite different from scalar algebra where ab=ba (that is 23=32). 8 7 Matrix Multiplication When we say matrix B is postmultiplied by matrix A, we mean BA. When we say matrix B is premultiplied by matrix A, we mean AB. 1 2 A Using: 3 4 2 B 1 1 3 Proof that the transpose of a product two matrices is the product of the transposes in reverse order,e.g. (AB)T=BTAT Positive integral powers of square matrices.. we define them as we did for scalars A2 = AA A3 = AAA etc. Could you realize why A has to be square for An (for n2) ? Only then does A have the same number of columns as the number of rows, necessary for matrix multiplication. One characteristic of the identity matrix.. If I is the unit matrix, and A is a square matrix of the same order, then I A = A I Could you realize why A has to be square matrix in this case? One definition.. A diagonal matrix is a square matrix whose off-diagonal elements, aij for ij, are all equal to zero. Example: An identity matrix is a diagonal matrix. Example... One illustration of how matrix multiplication my be used. A simple problem from the field of decision making. Example.. A certain fruit grower in Florida has a boxcar loaded with fruit ready to the shipped north. The load consists of 900 boxes of oranges, 700 boxes of grapefruit, and 400 boxes of tangerines. The market prices, per box, of the different types of fruit in various cities are given by the following chart. Example.. New York Cleveland St. Louis oranges $4 per box $5 per box $4 per box grapefruit $2 per box $1 per box $3 per box tangerines $3 per box $2 per box $2 per box Oklahoma City $3 per box $2 per box $5 per box To which city should the carload of fruit be sent in order for the grower to get maximum gross receipts for this fruit? Solution... Consider the chart.. oranges New York $4 per box Cleveland $5 per box St. Louis $4 per box Oklahoma City $3 per box grapefruit $2 per box $1 per box $3 per box $2 per box tangerines $3 per box $2 per box $2 per box $5 per box as the “price matrix”, and form the quantity matrix 900boxes 700boxes 400boxes Example.. The product of these matrices, as shown below, yields an “income matrix” where each entry represents the total income from all the fruit at the respective cities. 4 2 3 3600 1400 1200 6200 5 1 2 900 4500 700 800 6000 700 4 3 2 3600 2100 800 6500 400 3 2 5 2700 1400 2000 6100 Example.. 4 5 4 3 2 3 3600 1400 1200 6200 900 1 2 4500 700 800 6000 700 3 2 3600 2100 800 6500 400 2 5 2700 1400 2000 6100 The largest entry in the income matrix is 6500, and therefore the greatest income will come from St. Louis. Exercises.. Given V1 and V2. Find V1 V2. V1 3 0 6 2 V2 1 2 3 7 V1V2=3(-1)+02+6 3+2 7=29 Multiply... 2 A 4 0 3 and 1 2 3 0 6 4 B 1 2 5 3 9 6 27 1 A B 11 2 19 20 2 4 10 6 Matrix multiplication laws.. The associative law for multiplication is ABC=(AB)C which means that if we wish to find the product of three matrices, we can either multiply the first two matrices and then postmultiply it by the third matrix, or we can multiply the second and third matrices and then premultiply it by the first matrix. Matrix Multiplication The distributive law for multiplication with respect to addition requires that identical results be obtained whether you add the products AB and AC or multiply A by the sum of B and C: A(B+C)=AB+BC MSCI2400 Analytical Methods in Management Dr LAZO ROLJIC Fulbright Fellow DuPree College of Management Faculty of Economics University of Banja Luka Bosnia and Herzegovina ... -Sigma notation -Elementary matrix transformation -Determinants -Inverse matrix -Gauss-Jordan operations -Solution of simultaneous system of equations - Cramer’s rule Sigma notation When matrix multiplication is discussed in general, the co called “ notation” (read sigma notation) is very helpful. is letter from Greek alphabet and in mathematics usually stands for “sum of”. 5 For example, the sum 12 2 2 32 4 2 52 k 2 k 1 The expression is read “the sum of k2 where k ranges from 1 through 5”. k is called the index of summation. Examples: 50 2+4+6+8+10+…+98+100= 2k k 1 100 x1+x2+x3+…+x100= x k 1 k 3 ai1+ai2+ai3= a k 1 ik 3 a11b11+a12b21+a13b31= a1k bk 1 k 1 a11b11+a12b21+a13b31= 3 a k 1 b 1k k 1 This example my be recognized as the entry in the first row and first column of the product of the two matrices. a11 a21 am1 a12 a22 . . am 2 ... a1 p b11 b12 ... a2 p b21 b22 . . . . ... amp bp1 bp 2 ... b1n c11 c12 b ... 2 n c21 c22 . . . . ... bpn cm1 cm2 ... c1n ... c2 n . . ... cmn Elementary matrix transformation.. Definition: The three operations: 1) interchange any two rows, (this is equivalent to writing the equation in a different order and obviously this does not effect the solutions of the system), 2) multiply any row by a nonzero scalar, 3) add to any row a scalar multiple of another row, are called elementary row operations. Elementary Operations Elementary column operations are defined by replacing the word “row” by “column” throughout the preceding definition. An elementary operation is any operation that is either an elementary row operation or an elementary column operation. Equivalent matrices If a matrix A can be transformed into a matrix B by means of one or more elementary operation, we write A B and say that A is equivalent to B. 1 2 Example.. The matrix 3 3 4 1 Can be transformed to 1 3 4 0 9 9 by elementary row operation 3) - multiply the first row by 2 and add to second row (R2=2R1+R2). This then can be transformed to 1 3 4 0 1 1 by elementary operation 2) -multiply the second row by 1/9 (R2=R2/9). Exercises.. 2 4 1 5 , 2 0 1 3 R2=(-2R1+R2) Which elementary row operation (if any) transforms the first matrix into the second? 2 4 1 5 -2R1 2 4 1 5 +R2 -4 -2 Which elementary row operation (if any) transforms the first matrix into the second? 2 1 4 2 6 1 1 4 2 2 3 4 Determinants.. A matrix has no numerical value. However, every square matrix of scalars has what is called determinant. Determinants.. The Determinant is a number associated with every square matrix. The determinant of A, denoted by |A|, is obtained as the sum of all possible products in each of which there appears one and only one element from each row and each column of A. The determinant of A is.. a11 a12 ... a1n A am1 . . ... . . . ... . am2 ... amn Finding the determinant of a 2 by 2 matrix is straightforward.. 11 12 a a A a21 a22 + - a11 a12 A a11a22 a21a12 a21 a22 Determinants.. For a larger-order determinants, define Dij as the determinant of the [(n-1)x(n-1)] matrix formed by striking out (or deleting) the ith row and jth column of A. Dij is called the (i,j) “minor” of A. For example: minor D11 of |A|... a11 a12 a13 A a21 a22 a23 a31 a32 a33 a22 a23 D11 a32 a33 Next, we define the (i,j) cofactor of A to be ij=(-1)i+j Dij that is, the (i,j) minor ( Dij ) with a negative sign attached if the sum of subscripts is odd, and a positive sign if the sum is even. Examples: 11= (-1)2D11, 12=(-1)3D12, 13=14D13, etc. Determinants.. The minor Dij of the element aij is the determinant obtained from the square matrix A by striking out (or deleting) the ith row and jth column. So, a11 a12 a13 A a21 a22 a23 a31 a32 a33 + + a11 a12 A a 21 a 22 a31 a32 a13 a 22 a 23 a11 a32 a33 a 23 a 21 a12 a33 a31 a 23 a 21 a13 a33 a31 a 22 a32 Example.. 11 4 6 A11 (1) 3 2 3 A 2 1 5 4 3 2 6 A12 (1) 1 2 1 2 7 6 2 A13 (1) 1 3 2 4 1 3 4 6 2 6 2 4 A 3 5 7 3(10) 5(10) 7(10) 10 3 2 1 2 1 3 Properties of determinants 1) If a matrix B is formed from a matrix A by the interchange of two parallel lines (rows or columns) then |A|=-|B|. 2) The determinants of a matrix and its transpose are equal; that is, |A|=|AT|. 3) If all of the entries of any row or column of matrix A are zero, then |A|=0. 4) The determinant of a matrix with two identical parallel rows or columns is zero. Example for properties of 3) and 4) 0 1 4 2 2 1 A 2 1 6 0 A 3 3 1 0 0 0 0 4 4 0 Inverse matrices... Inverse matrix.. The inverse of a matrix A of order (n x n) is the matrix A-1 such that AA-1=I, or A-1A=I. In other words, the matrix which when multiplied by the matrix A produces the identity matrix is said to be inverse of A and is denoted by A-1. Since A conforms to A-1 and A-1 conforms to A, A must be square. Therefore, we conclude that only square matrices (but not all, f.e. singular) have inverses. The singular matrix The singular matrix is matrix which determinant is equal zero. Inverse Matrices Division is not defined in matrix algebra. The matrix algebra counterpart of division in scalar algebra is achieved through matrix inversion. In algebra of real numbers, the inverse of of X is 1/X, is a quantity which when multiplied by X yields 1. Inverse Matrix The inverse of a square matrix A, which we designate A-1, is a matrix which when postmultiplied or premultiplied by A yields the unit or identity matrix. AA-1=I, A-1A=I. Or, we can say, if two square matrices A and B, each of degree p, satisfy the equation AB=Ip we call B the inverse of A, written B=A-1. Proof.. This is valid AA-1=I, A-1A=I. Suppose is given a matrix B which satisfies BA=I. If we now multiply each side of the last equation by A-1 on the right, we have BAA-1=IA-1 which reduces to B=A-1. Matrix Inverses Thus, the inverse of matrix A is.. 1 2 3 1 1 / 3 1 / 9 5 / 9 1 A 0 1 2 2 / 3 1 / 9 4 / 9 3 2 1 1 / 3 5 / 9 2 / 9 How we calculate A-1? a11 a12 A a21 a22 The inverse of any square matrix A consisting of two rows and two columns is computed as follows: a / d a / d 22 12 1 A a21 / d a11 / d Where d=a11a22-a21a12 is determinant of the 2x2 matrix A. Example.. 0.7 0.3 A 0.3 0.9 Then d=0.70.9-(-0.3)(0.3)=0.54 and 0 . 9 / 054 . 0 . 3 / 054 . 167 . 056 . 1 A . 0.7 / 054 . 056 . 130 . 0.3 / 054 Gauss-Jordan operations.. One way to form the inverse of a square matrix is to perform simultaneously on a unit matrix, of the same order, necessary row operations to convert the left matrix to I. This is known as Gauss-Jordan operations. Thus, given 2 4 A 3 2 Gauss-Jordan operations.. Arrange alongside A a unit matrix I: 2 4 1 0 3 2 0 1 2 4 1 0 3 2 0 1 Now perform on A the row-column operations via which the Gauss-Jordan procedure is employed to convert A to a unit matrix-performing these operations simultaneously on I. Thus, when the first row is divided by 2 to yield a 1 in the first row, first column of A, divide the first row of I by 2 also (R1=R1/2). When the second row of A is replaced by the sum of itself and the new first row multiplied by -3, replace the second row of I similarly (R2=R2+(-3)R1). January 21, 1999 Top Ten Ways to Succeed in MSCI 2400 .Understand –Don’t Memorize . Ask when Something don’t Understand .Don’t Miss Class .Don’t Fall Behind .Complete all Homework Individually .Read the Text .Study the Homework .Study the Handouts . Study the old Exams .Participate and Thoroughly Understand Assignments .Come to Office Hours for Help Gauss-Jordan operations.. The final result is 1 0 IA 1 0 1 / 4 1 / 2 1 3 / 8 1 / 4 Thus, as A is transformed into I, I is transformed into A-1. Why is the inverse matrix so important? To get some appreciation and understanding of the utility of the matrix inverse, consider the following linear system of equations. Let 15 . 1 2.4 A 1 5 1 . 3 3.5 15 x1 X x2 x3 Thus if we form AX=b 2000 b 8000 5000 15 . 1 2.4 A 1 5 1 . 3 3.5 15 x1 X x2 x3 2000 b 8000 5000 we really have the following system of three equations in three unknowns: 1.5x1+ x2+ 2.4x3=2000 x1+5x2+ x3=8000 1.5x1+3x2+3.5x3=5000 To obtain a solution to this system when written in the matrix form, we start with the equation AX=b AX=b Multiplying both sides of the equation on the left by A-1, we obtain A-1AX=A-1b Evaluating the left-hand side by using the associative law, we have A-1AX=(A-1A)X=IX=X and thus our solution is X=A-1b According to X=A-1b, then, to solve m equations in m unknowns, we should find the inverse of the matrix of coefficients and multiply this by the column vector of constants b.This result will be a columnvector X. Thus, if we are able to find the inverse of the matrix of coefficients, it is easy to find the solution of the system of equations. 15 x1 . 1 2.4 A 1 5 1 X x2 . 3 3.5 15 x3 2000 b 8000 5000 AX=b, according to X=A-1b, we should find the inverse of the matrix A and multiply this by the column vector b. The result will be a column vector X. 15 . 1 2.4 1 1 5 1 0 . 3 3.5 15 0 0 1 0 0 0 1 Now we should to perform next elementary row-column operations to get A-1: 15 . 1 2.4 1 0 0 1 5 1 0 1 0 15 . 3 35 . 0 0 1 R1=R1/1.5 R2=R2-R1 16201 . 0.4134 12291 . 1 R3=R3-1.5R1 A 0.2235 01844 . 01006 . R2= R2/4.333 05028 . 0.3352 0.7263 R3= R3/1.37692 R2=0.13846R3+ R2 R1=-1.69231R3+ R1 After that, using X=A-1b we will calculate solution vector X. 16201 . 0.4134 12291 . 2000 4019 . 1 X A b 0.2235 01844 . 01006 . . 8000 15319 05028 . 0.3352 0.7263 5000 557 . The best computational scheme for calculating of a solution of n equations with n variables (vector Xp) is to follow a procedure similar to this described earlier for finding the inverse matrix. But, instead of computing the inverse matrix, we may set up together A and b and then apply the necessary row transformations to reduce A to its identity form. A b I A-1b In other words, by applying the same transformations to b that we did to A, we have converted b into A-1b which is solution vector. In this manner we compute A-1b without actually finding A-1 and without performing the multiplication of A-1 by b. If for any reason we should want the inverse of A, we can get it by setting up I behind A on this manner: A I b After the proper row operations, we will have I A-1 A-1b Inverse matrix.. Note that all matrices do not have inverses, yet whenever the inverse to a matrix can be found, it is unique. But, the existence of a solution is not dependent on the existence of the inverse matrix. Reasons for that almost are: redundancy, inconsistency or singularity of the set of equations. Redundancy Two or more equations of the set may be linearly dependent (that is, a combination of sums or differences of multiples of other equations). The dependent equations are redundant and can be omitted from the system. We then have a system of fewer than m equations in m variables. (Many solutions). Example of one redundant system.. 2x1 -4x2 + x3=7 -x1+7x2 -3x3=12 4x1 -8x2+2x3=14 Inconsistency On the other hand, system can be inconsistent and then A-1 does not exist. For example, consider: 3x1+2x2+ x3=14 2x1 -x2+3x3=10 5x1+ x2 +4x3=5 Here the third equation is inconsistent with the first two and no solution exist. Singularity Finally, a matrix could be singular and its inverse does not exist. 3x1+3x2+ x3=14 2x1+2x2+2x3=10 5x1+5x2 -2x3 =7 x1 and x2 have equal coefficients in each equation and, hence, are dependent on each other. The matrix of coefficients is then singular and its inverse does not exist. Note! Every nonsquare matrix is singular. Exercise.. Compute the inverse matrix of A: 4 8 A 6 4 R1/4; (-6R1+R2); R2/8; (-2R2+R1) 4 8 A 6 4 First set the problem in the following form: 4 6 1 6 A I 8 4 1 0 0 1 2 4 1 / 4 0 0 1 1 2 1/ 4 0 0 8 3 / 2 1 Now perform the G-J steps on both of these matrices simultaneously. (First R1/4) Now multiply R1 by -6 and add result to the second row. (-6R1+R2) Next divide R2 by -8. (R2/-8) 1 2 1/ 4 0 0 8 3 / 2 1 1 2 1 / 4 0 0 1 3 / 16 1 / 8 I A-1 1 0 1 / 8 1 / 4 0 1 3 / 16 1 / 8 Finally, (-2R2+R1) Solution of a set of simultaneous linear equations Any set of simultaneous linear equations has a convenient representation using matrix notation. The system a11x1+a12x2+…+a1nxn = b1 a21x1+a22x2+…+a2nxn = b2 … … … … … … am1x1+am2x2+…+amnxn = bm can be written as AX=b where.. A=[aij](m,n) x1 . X . . x n b1 . b . . b m If A is square (m=n) and nonsingular, the solution vector is given: 1) by X=A-1b (using matrix inverse notation) or 2) using Cramer’s rule for nonsingular determinants i det( A) xj , j 1,2,..., n det A where (iA) denote the matrix obtained from A by replacing the jth column of A by the vector b. Example of Cramer’s rule application.. 2x1+x2+ x3=0 x1 -x2+5x3=0 x2 - x3=4 x2 x1 0 1 1 0 1 5 4 1 1 2 1 1 1 1 5 0 1 1 2 0 1 2 1 0 1 0 5 1 1 0 0 4 1 0 1 4 2 1 1 2 1 1 1 1 5 1 1 5 0 1 1 0 1 1 x3 Homework 1 January 19,1999 1) Express the product of 2 by 2 matrices using notation. 2) Write the following without notation: 5 k k 1 7 ( k 2) k 3 4 a k 1 k 3 a k 1 2k ak 3 3) Express the sums in sigma notation: a11 a12 b11 b12 a 21 a22 b21 b22 b13 b23 4) Find by definition determinant of the matrix: 2 A 4 0 1 8 7 3 6 5 5) Using matrix A given below proof that the determinants of a matrix and its transpose are equal. 1 0 0 2 2 0 3 1 4 6) Using elementary matrix operation suggested below (or self selected), find the solution (vector X) of the next system: 15 . 1 15 . 1 5 3 2.4 x1 2000 1 x 2 8000 3.5 x 3 5000 R1=R1/1.5 R2=R2-R1 R3=R3-1.5R1 R2= R2/4.333 R3= R3/1.37692 R2=0.13846R3+ R2 R1=-1.69231R3+ R1 7) Using Cramer’s rule find solution of next system of simultaneous equations: 2x1+x2+ x3=0 x1 -x2+5x3=0 x2 - x3=4