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For parts (a) to (c) of this question, we consider the 2 × 3 matrix A = 1 1 0 011 9). (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = -X(X - 1)(A-3)

For parts (a) to (c) of this question, we consider the 2 × 3 matrix A = 1 1 0 011 9). (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = -X(X - 1)(A-3)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = (619) 0 (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = X(X - 1)(A-3) (b) State the eigenvalues of ATA and find the eigenspace associated with each eigenvalue. Hence determine if AT A is diagonalisable. (c) We can use the eigenvalues and eigenvectors of ATA to express A = UEVT as the multi- plication of three matrices U, E and V. 01 0 . The matrix > = is a "diagonal" matrix consisting of oi's called the 0 02 singular values. The values o, are formed by the square root of the positive eigenvalues of AT A ordered in decreasing order. The matrix V consists of columns vectors ; called right singular vectors of A. The vectors 's are eigenvectors of AT A normalised so that ||||=1 with respect to the standard inner product on R³. The ordering of v; should be the same as that for ₁. • The matrix U consists of columns vectors u; called left singular vectors of A. If defined, the vectors uj can be extracted by the identity uj = Auj. (Note: In the next part, we will show that this identity is applicable for a more general m x n matrix). Evaluate the matrices U, E, and V for A and validate that A = UEVT.
Transcribed Image Text:For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = (619) 0 (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = X(X - 1)(A-3) (b) State the eigenvalues of ATA and find the eigenspace associated with each eigenvalue. Hence determine if AT A is diagonalisable. (c) We can use the eigenvalues and eigenvectors of ATA to express A = UEVT as the multi- plication of three matrices U, E and V. 01 0 . The matrix > = is a "diagonal" matrix consisting of oi's called the 0 02 singular values. The values o, are formed by the square root of the positive eigenvalues of AT A ordered in decreasing order. The matrix V consists of columns vectors ; called right singular vectors of A. The vectors 's are eigenvectors of AT A normalised so that ||||=1 with respect to the standard inner product on R³. The ordering of v; should be the same as that for ₁. • The matrix U consists of columns vectors u; called left singular vectors of A. If defined, the vectors uj can be extracted by the identity uj = Auj. (Note: In the next part, we will show that this identity is applicable for a more general m x n matrix). Evaluate the matrices U, E, and V for A and validate that A = UEVT.
(d) As promised, we will explore the identity for extracting U. Suppose we start with a fixed, but unknown mxn matrix P with m <n and a SVD decomposition given by P = UEVT. Show that the following identity holds for valid choices of j. Púj = ojuj For this part, we may assume without proof that the eigenvectors of PTP forms an or- thogonal basis of R
Transcribed Image Text:(d) As promised, we will explore the identity for extracting U. Suppose we start with a fixed, but unknown mxn matrix P with m <n and a SVD decomposition given by P = UEVT. Show that the following identity holds for valid choices of j. Púj = ojuj For this part, we may assume without proof that the eigenvectors of PTP forms an or- thogonal basis of R
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