For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = (19) 01 (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = X(X - 1)(x-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, Σ and V. 0 The matrix > = 8) is a "diagonal" matrix consisting of o'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 i 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 ₂. . The matrix U consists of columns vectors u; called left singular vectors of A. If defined, the vectors u can be extracted by the identity uj Av. (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.

For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = (19) 01 (a) Compute AT A and show that its characteristic polynomial can be factored as: XATA(X) = X(X - 1)(x-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, Σ and V. 0 The matrix > = 8) is a "diagonal" matrix consisting of o'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 i 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 ₂. . The matrix U consists of columns vectors u; called left singular vectors of A. If defined, the vectors u can be extracted by the identity uj Av. (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.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 29EQ

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