![Least Squares Adjustment](http://s1.studyres.com/store/data/010447319_1-4ded460f2465c6512f4fbd753800c5de-300x300.png)
Sept. 3, 2013 Math 3312 sec 003 Fall 2013
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
Basic Matrix Operations
... A matrix is a rectangular or square grid of numbers arranged into rows and columns. Each number in the matrix is called an element, and they are arranged in what is called an array. The plural of “matrix” is “matrices”. Matrices are often used in algebra to solve for unknown values in linear equatio ...
... A matrix is a rectangular or square grid of numbers arranged into rows and columns. Each number in the matrix is called an element, and they are arranged in what is called an array. The plural of “matrix” is “matrices”. Matrices are often used in algebra to solve for unknown values in linear equatio ...
Properties of Matrix Operations - KSU Web Home
... justify your answer. When a statement is true, you can justify your answer by either explaining why it is true, or citing a theorem or property. If the statement is false, give a counter example. In addition, if the statement is false, try to see what condition could be added to make it true. (a) Ev ...
... justify your answer. When a statement is true, you can justify your answer by either explaining why it is true, or citing a theorem or property. If the statement is false, give a counter example. In addition, if the statement is false, try to see what condition could be added to make it true. (a) Ev ...
Notes
... The difference 2 CC is positive semi definite because 2 wCC w ( C w)( Cw) is the non-negative squared length of the vector Cw (inner product). We use definiteness in evaluation of minima and maxima; e.g. for maximizatior of utility the Hessian should be negative definite. If a matr ...
... The difference 2 CC is positive semi definite because 2 wCC w ( C w)( Cw) is the non-negative squared length of the vector Cw (inner product). We use definiteness in evaluation of minima and maxima; e.g. for maximizatior of utility the Hessian should be negative definite. If a matr ...
Matrix
... Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions Dimensions are the rows x columns ...
... Matrix: is any rectangular array of numbers written within brackets; represented by a capital letter; classified by its dimensions Dimensions are the rows x columns ...
54 Quiz 3 Solutions GSI: Morgan Weiler Problem 0 (1 pt/ea). (a
... product of elementary matrices (take the product of the inverses of the elementary matrices used to reduce A to the identity). (d). If det A = 3 then what is det A−1 ? Solution: Since (det A)(det A−1 ) = det(I) = 1, det A−1 = ...
... product of elementary matrices (take the product of the inverses of the elementary matrices used to reduce A to the identity). (d). If det A = 3 then what is det A−1 ? Solution: Since (det A)(det A−1 ) = det(I) = 1, det A−1 = ...
PDF
... 5. Let O denote the n × n matrix with all entries equal to 0. Suppose that A and B are n × n matrices with AB = O , but B 6= O . Show that A cannot be invertible. (Hint: suppose it is: what does that tell you about B ?) If A has an inverse A−1 , then A−1 A = In , the n × n identity matrix. But then ...
... 5. Let O denote the n × n matrix with all entries equal to 0. Suppose that A and B are n × n matrices with AB = O , but B 6= O . Show that A cannot be invertible. (Hint: suppose it is: what does that tell you about B ?) If A has an inverse A−1 , then A−1 A = In , the n × n identity matrix. But then ...
Appendix 4.2: Hermitian Matrices r r r r r r r r r r r r r r r r r r
... A square n×n matrix B is said to be Hermitian if B* = B. Here, the * denotes complexconjugate transpose (some authors use an “H” as a subscript to denote complex-conjugate transpose, and they would write BH = B). We need two important attributes of Hermitian matrices. First, their eigenvalues are al ...
... A square n×n matrix B is said to be Hermitian if B* = B. Here, the * denotes complexconjugate transpose (some authors use an “H” as a subscript to denote complex-conjugate transpose, and they would write BH = B). We need two important attributes of Hermitian matrices. First, their eigenvalues are al ...
MAT1001, Fall 2011 Oblig 1
... You may work on these exercises together with your fellow students, but each of you has to hand in his/her own version of the solution to the assignment for approvement. Please put your name and course code (MAT 1001) on your answer sheet. The assignments must be handed in to the Mathematics Departm ...
... You may work on these exercises together with your fellow students, but each of you has to hand in his/her own version of the solution to the assignment for approvement. Please put your name and course code (MAT 1001) on your answer sheet. The assignments must be handed in to the Mathematics Departm ...
Slide 1
... Solving for the eigenvalues involves solving det(A–λI)=0 where det(A) is the determinant of the matrix A For a 2x2 matrix the determinant is a quadratic equation, for larger matrices the equation increases in complexity so lets solve a 2x2 Some interesting properties of eigenvalues are that the prod ...
... Solving for the eigenvalues involves solving det(A–λI)=0 where det(A) is the determinant of the matrix A For a 2x2 matrix the determinant is a quadratic equation, for larger matrices the equation increases in complexity so lets solve a 2x2 Some interesting properties of eigenvalues are that the prod ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.