Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
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.

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
See similar textbooks
icon
Related questions
Question

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. 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.
Transcribed Image Text: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.
Expert Solution
steps

Step by step

Solved in 6 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS