2.c) 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.0The matrix > =8) is a "diagonal" matrix consisting of o's called the0 02singular values. The values o, are formed by the square root of the positive eigenvaluesof AT A ordered in decreasing order.The matrix V consists of columns vectors i called right singular vectors of A. Thevectors 's are eigenvectors of AT A normalised so that |||| = 1 with respect to thestandard 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 ujAv. (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.

(c) Given: The 2×2 matrix A=110011 To do: We need to evaluate the matrices U, ∑ and V for A and…

1 For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = 0 1 (a) Compute AT A and show that its characteristic polynomial can be factored as: 1 0 1 XATA(X) = -X(X − 1)(x − 3) - (b) State the eigenvalues of ATA and find the eigenspace associated with each eigenval Hence determine if AT A is diagonalisable. (c) We can use the eigenvalues and eigenvectors of ATA to express A = UEVT as the mu plication of three matrices U, Σ and V. 00 The matrix Σ = (90) is a "diagonal" matrix consisting of oi's called singular values. The values o; are formed by the square root of the positive eigenval of AT A ordered in decreasing order. . The matrix V consists of columns vectors ; called right singular vectors of A. T vectors 's are eigenvectors of AT A normalised so that ||||=1 with respect to standard inner product on R³. The ordering of v; should be the same as that for • The matrix U consists of columns vectors u; called left singular vectors of A. If defin the vectors ; can be extracted by the identity uj = Av. (Note: In the next pa 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.

1 For parts (a) to (c) of this question, we consider the 2 x 3 matrix A = 0 1 (a) Compute AT A and show that its characteristic polynomial can be factored as: 1 0 1 XATA(X) = -X(X − 1)(x − 3) - (b) State the eigenvalues of ATA and find the eigenspace associated with each eigenval Hence determine if AT A is diagonalisable. (c) We can use the eigenvalues and eigenvectors of ATA to express A = UEVT as the mu plication of three matrices U, Σ and V. 00 The matrix Σ = (90) is a "diagonal" matrix consisting of oi's called singular values. The values o; are formed by the square root of the positive eigenval of AT A ordered in decreasing order. . The matrix V consists of columns vectors ; called right singular vectors of A. T vectors 's are eigenvectors of AT A normalised so that ||||=1 with respect to standard inner product on R³. The ordering of v; should be the same as that for • The matrix U consists of columns vectors u; called left singular vectors of A. If defin the vectors ; can be extracted by the identity uj = Av. (Note: In the next pa 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.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

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