Download Sections 3.1-3.2

Chapter 3 Outline
Section 3.1 Matrices: Sums and Products
Definition of a Matrix: A matrix is a rectangular array of elements or entries (numbers
or functions taking numerical values) arranged in rows (horizontal) and (vertical).
a1n 
 a11

 . We say that matrix A has m rows and n columns or is of order
A

a

a
mn 
 m1
m n (read as “m by n”). Usually we usually denote matrices by boldface capital letters.
Note the entries are denoted with lowercase letter with subscripts (row first, then
column).
An m1 matrix is called a column vector and a 1 n is called a row vector.
Equal Matrices: Two matrices of the same order are equal if their corresponding entries
are equal. If matrices A   aij  and B  bij  are both m n , then A  B if and only if
aij  bij , 1  i  m , 1  j  n .
Special Matrices
The m n zero matrix has all its entries equal to zero. It is denoted by 0 mn or if it is
clear from the context just by 0.
The matrix entries aij in an m n matrix for which i  j are called diagonal elements; all
of them make up the (main) diagonal of a matrix.
A diagonal matrix is a square matrix for which all the nondiagonal elements are zero.
The diagonal elements may be nonzero or zero.
An n  n identity matrix is a diagonal matrix that all its diagonal elements equal to one.
It is denoted by In (or just I if the order is clear from the context.).
If we flip a matrix “diagonally” so that the rows become columns and the columns
become rows, we get a new matrix called the transpose of the original matrix. We write
A T for the transpose of A.
Properties of Transposes
Let A and B be matrices of compatible orders for the indicated sums and products.
1) (AT )T = A
2) (A  B)T = AT  BT
3) For any scalar k, (kA)T  kAT
4) ( AB)T = BT AT
Symmetric matrix is a matrix in which (AT ) = A
Matrices with Function Entries
Matrices with entries that are functions rather than constants are important because they
help us deal with differential equations and their solutions.
 x1 (t ) 
a1n (t ) 
 a11 (t )


x
(
t
)
2



x (t )  
and
A(t )  



 a (t )

a
(
t
)


mn
 m1

x
(
t
)
 n 
We say that matrices x(t) and A(t) are continuous, piecewise continuous, or differentiable
providing that every entry has the required property. The derivatives of a matrix of
function is the matrix of the derivatives of the entries.
Matrix Arithmetic
Matrix Addition:
Two matrices of the same order are added (or subtracted) by adding (or subtracting)
corresponding entries and recording the results in a matrix of the same size. Using matrix
notation, if A   aij  and B  bij  are both m n ,
A + B = a ij   bij    a ij  bij 
A - B = a ij   bij   a ij  bij 
Multiplication by a Scalar: To find the product of a matrix and a number, real or
complex, multiply each entry of the matrix by that number. This is called multiplication
by a scalar. Using matrix notation, if A   aij  , then cA  caij    aij c   Ac
Properties of Matrix Addition and Scalar Multiplication:
Suppose A, B, and C are m n matrices and c and k are scalars. Then
 AB = BA
(Commutativity)
 A  (B + C) = (A + B) + C
(Associativity)
 c(kA) = (ck ) A
(Associativity)
 A0 = A
(Zero Element)
 A  ( A)  0 where –A denotes (-1) A
(Identity Element)
 c( A  B) = cA + cB
(Distributivity)
 (c  k ) A = cA + kA
(Distributivity)
Matrix Multiplication
Matrix multiplication is based on “multiplying” a row vector and a column vector (of
equal length) element by element and adding the products together. Before looking at the
product of two matrices, first we introduce the idea of a scalar product of two vectors.
Scalar Product: The scalar product (or dot product) of two vectors, a row vector and a
column vector of equal length n, is the result of adding the products of the corresponding
entries. For row vector a and column vector b, the scalar product denoted a  b , is
 b1 
b 
a  b = [ a1 a2
an ]   2 
 
 
bn 
Matrix Product: Let A be an m  r matrix and B be an r  n . The ijth entry of C=AB is
the dot product of the ith row vector of A and the jth column vector of B:
 b1 j 
b 
2j
cij = [ai1 ai 2
ain ]   
 
 
 bnj 
 ai1b1 j  ai 2b2 j 
n
 ainbnj   aik bkj .
k 1
The product C has order m n .
Properties of Matrix Multiplication
 A(BC) = (AB)C
(Associativity)
 A(B + C) = AB + AC
(Distributivity)
 (B + C)A = BA + CA
(Distributivity)
 In general, the matrix product is not commutative.
Except in special cases AB  BA
Identity matrices behave rather like the number 1, and zero matrices behave rather like
the number 0. For an m n matrix A,
 AI n  A ,
Im A  A
 A0np  0mp , 0qm A  0qn , for any p and q
Matrix Differentiation Rules:
For differentiable matrices A and B (whose entries are functions) and scalar constant c,
 ( A + B) = A + B
 (cA) = cA
 ( AB) = AB + AB
Vector Addition and Scalar Multiplication:
 x1 
 y1 
x 
y 
2

 2
x
=
y
=
Let
and
 
 
 
 
 xn 
 yn 
Be vectors in n and c be any scalar. Then x + y and cx are defined, respectively, as
 x1   y1   x1  y1 
 x1   cx1 
x   y  x  y 
 x   cx 
2
 2   2   2
c 2   2
and
    

   
    

   
 xn   yn   xn  yn 
 xn   cxn 
Properties of Vector Addition and Scalar Multiplication:
Suppose u, v and w are vectors in n and c and k are scalars. Then
 u v = v u
(Commutativity)
 u  ( v + w) = (u + v) + w
(Associativity)
 c(ku) = (ck )u
(Associativity)
 u0 = u
(Zero Element)
 u  (u)  0
(Identity Element)
 c(u  v) = cu + cv
(Distributivity)
 (c  k )u = cu + ku
(Distributivity)
The General Scalar Product: Let x, y 
y =  y1
n
yn  . Then
y2
x • y = x1 y1  x2 y2 
with x =  x1
x2
xn  and
n
 xn yn   xk yk
k 1
and is scalar.
Definition
Two vectors is
n
are called orthogonal when there scalar product is zero.
Definition
For any vector v  n , the length or absolute value of v is a nonnegative scalar denoted
by v and defined to be v  v • v .
Systems of Differential Equations
Systems of DEs can often be written using matrix-vector notation.
a) If a system of DES is autonomous and linear,
x  ax  by,
y  cx  dy,
 x
a b 
it can be written as x = Ax , where x    and A  
.
 y
c d 
b) A nonautonomous system of DES
x  7 x  4 y  t ,
y  2 x  y  et ,
can be written in “matrix-vector form” as x = Ax + f (t ) ,
 x
 7 4
 t 
where x    , A  
,
and
f
(
t
)


 e t  .
 y
 2 1 
 
Notice that the in the equation the right-hand side is separated into homogeneous
and nonhomogeneous parts of a linear system.
c) Another instructive example
x  1  x  y ,
y  x  y 2 ,
Shows how far matrix notation can get you in a nonlinear system. The system can be
written in matrix-vector form as x = Bx + g (x)
 x
 1 1
 1 
with x    , B  
, and g(x)   2  .

 y
1 0
 y 
Notice that in the equation we have separated the right-hand side into its linear and
nonlinear parts.
Section 3.2 Systems of Linear Equations
A m n system of linear equations is a set of m equations in n variables x1 , x2 ,
of the form
a11 x1  a12 x2   a1n xn  b1 ,
a21 x1  a22 x2 
 a2 n xn  b2 ,
am1 x1  am 2 x2 
 amn xn  bm
, xn
where aij and bi are constants and the xi are the unknown variables. If the bi are all
equal to zero, the system is homogeneous. A solution is a point in
coordinates satisfy the system of equations.
n
whose
The system can be written as
a1n   x1   b1 
 a11 a12
a
a2 n   x2   b2 
 21 a22


   

   
amn   xn  bm 
 am1 am 2
with the compact matrix-vector form Ax  b .
The system is homogeneous if and only if b is the zero vector.
Elementary Row Operations
There are three operations that can be performed on a row of matrix that alters the
row but applying the manipulations will change the original system into an
equivalent system, a system with the same set of solutions.
The matrix to which the operations will be applied is called the augmented matrix of
the system of Ax  b . It is formed by adding the entries of the column vector b to
those of the coefficient matrix A, creating a matrix of order m  (n  1) . The
augmented matrix of system above is
 a11 a12
a1n b1 


a21 a22
a2 n b2 

 A b  
.




amn bm 
 am1 am 2
The elementary row operations outlined below both describe in words and by a
shorthand notation in which Ri denotes the ith row of the matrix before the operation
is applied and Ri* represents the ith row of the matrix after the operation is carried out.
Elementary Row Operations:
 Interchange row i and row j:
Ri  R j
(or Ri*  R j , R*j  Ri ).


Multiply row i by a constant c  0 :
Ri*  cRi .
Add c times row j to row i (leaving row j unchanged):
Ri*  Ri  cR j .
Applied to the equations corresponding to the rows, these operations represent
interchanging the order of the equations, multiplying an equation by a nonzero constant
or adding a multiple of one equation to another equation.
The goal is for the final augmented matrix is to be put in reduced row echelon form
(RREF). RREF will produce an equivalent system from which the solution can be read
immediately.
Reduced Row Echelon Form (RREF): A matrix is in reduced row echelon form if the
following conditions are satisfied
i)
Zero rows are left at the bottom.
ii)
The leftmost nonzero entry of each nonzero row, called its pivot equals 1.
iii)
Each pivot is further to the right than the pivot in the row above it.
iv)
Each pivot is the only nonzero entry in its column.
A column of any matrix A is called a pivot column if it corresponds to a column with a
leading 1 in its RREF.
Gauss-Jordan Elimination
The Gauss-Jordan Elimination is the strategy that can be applied to any given matrix to
transform the matrix into RREF. (This procedure always works and the result will
always be the same. RREF is unique). Yet it is prone to arithmetic errors because the
operations are tedious. So be as careful as you can be on the exam and in the hws!
Gauss-Jordan Elimination:
The following procedure will solve the linear system Ax  b .
Step 1. Form the augmented matrix  A b  .
Step 2. Transform the augmented matrix to reduced row echelon form (RREF) using the
three basic elementary row operations.
Step 3. The linear system that corresponds to the matrix in reduced row echelon form,
which was obtained in Step 2, has exactly the same solutions as the given linear system.
For each nonzero row of the matrix in RREF solve for the unknown that corresponds to
the leading 1 in the row. The rows consisting of all zeros can be ignored because the
corresponding equation is satisfied for any values of the variables.
Existence and Uniqueness and the RREF
When an augmented matrix  A b  is in RREF, we can inspect it for answers to our initial
questions corresponding the existence and uniqueness of solutions to the linear system
Ax  b .
1. Existence: Are there any solutions? If there is a row in the RREF that is of the
form [0 0 k ] where k is nonzero, then the system has no solutions. (Such a row
would require 0 x1  0 x2   0 xn  k  0 , which is impossible.)
2. Uniqueness: Is there a solution, is it the only one? If the system is consistent
and every column in the RREF is a pivot column, then there is only one solution.
If instead there are some non-pivot columns, then there are infinitely many
solutions.
3. How do we calculate solutions? We can obtain these solutions by solving for
each variable xi that corresponds to a pivot column.
Terminology
The variables xi that correspond to a pivot column are called basic or leading variables.
If we must solve them in terms of the remaining variables xi , which correspond to the
non-pivot columns, then there are infinitely many solutions. The variables corresponding
to the non-pivot columns are called the free variables. They act as parameters and can
be chosen to be any real number.
Example 5 on p. 136-137 illustrates how to write a family of solutions when one or more
variables fail to correspond to a pivot column (the free variables). One would typically
be required to do this on exams or hws.
In a consistent system nonunique solutions arise when one or more variables fail to
correspond to a pivot column. If the system has n variables with r pivot columns, then
there will be n-r free variables and the solutions will form an (n-r)-parameter family. The
number r is called the rank of the matrix. If r is equal to the number of variables, then
there is a unique solution. If r is less than the number of variables, solutions are not
unique.