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We can use the eigenvalues and eigenvectors of ATA to express A = UΣVT as the multi- plication of three matrices U, Σ and V. The matrix > = (¹/20) 8) is a “diagonal" matrix consisting of oi's called the 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 vi's are eigenvectors of AT A normalised so that ||ū;|| = 1 with respect to the standard inner product on R³. The ordering of 7; should be the same as that for σ¿. • The matrix U consists of columns vectors ū¡ called left singular vectors of A. If defined, the vectors ū¡ can be extracted by the identity u¡ = Au¡. (Note: In the next part, we will show that this identity is applicable for a more general m × n matrix). Evaluate the matrices U, E, and V for A and validate that A = UEVT. o j

We can use the eigenvalues and eigenvectors of ATA to express A = UΣVT as the multi- plication of three matrices U, Σ and V. The matrix > = (¹/20) 8) is a “diagonal" matrix consisting of oi's called the 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 vi's are eigenvectors of AT A normalised so that ||ū;|| = 1 with respect to the standard inner product on R³. The ordering of 7; should be the same as that for σ¿. • The matrix U consists of columns vectors ū¡ called left singular vectors of A. If defined, the vectors ū¡ can be extracted by the identity u¡ = Au¡. (Note: In the next part, we will show that this identity is applicable for a more general m × n matrix). Evaluate the matrices U, E, and V for A and validate that A = UEVT. o j

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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We can use the eigenvalues and eigenvectors of ATA to express A = UΣVT as the multi- plication of three matrices U, Σ and V. The matrix Σ = 8) is a "diagonal" matrix consisting of oi's called the singular values. The values o; are formed by the square root of the positive eigenvalues of AT A ordered in decreasing order. 01 0 0 02 = • The matrix V consists of columns vectors ; called right singular vectors of A. The vectors ;'s are eigenvectors of AT A normalised so that ||vi|| 1 with respect to the standard inner product on R³. The ordering of 7; should be the same as that for σ¿. • 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 u¡j Av. (Note: In the next part, we will show that this identity is applicable for a more general m × n matrix). Evaluate the matrices U, Σ, and V for A and validate that A = UΣVT. = σj
Transcribed Image Text:We can use the eigenvalues and eigenvectors of ATA to express A = UΣVT as the multi- plication of three matrices U, Σ and V. The matrix Σ = 8) is a "diagonal" matrix consisting of oi's called the singular values. The values o; are formed by the square root of the positive eigenvalues of AT A ordered in decreasing order. 01 0 0 02 = • The matrix V consists of columns vectors ; called right singular vectors of A. The vectors ;'s are eigenvectors of AT A normalised so that ||vi|| 1 with respect to the standard inner product on R³. The ordering of 7; should be the same as that for σ¿. • 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 u¡j Av. (Note: In the next part, we will show that this identity is applicable for a more general m × n matrix). Evaluate the matrices U, Σ, and V for A and validate that A = UΣVT. = σj
0 = (19) 0 A =
Transcribed Image Text:0 = (19) 0 A =
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