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
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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
Transcribed Image Text:0
= (19)
0
A =
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