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## Homework Statement

I want to do a singular value decomposition for the following matrix:

[itex]M = \left\lceil 1 \ \ -1 \right\rceil[/itex]

[itex]\ \ \ \ \ \ \ \:\:\left\lfloor 1 \ \ -1 \right\rfloor[/itex]

## Homework Equations

[itex]M=U\Sigma V^{\ T}[/itex]

[itex]M^{\ T}M[/itex]

[itex]MM^{\ T}[/itex]

## The Attempt at a Solution

To determine the singular values for [itex]\Sigma[/itex], I first determined the eigenvalues from [itex]M^{\ T}M[/itex] (I could have also done it from [itex]MM^{\ T}[/itex]). The eigenvalues are 4 and 0, so the singular values are 2 and 0. So I end up with the following matrix:

[itex]\Sigma = \left\lceil 2 \ \ 0 \right\rceil[/itex]

[itex]\ \ \ \ \ \ \ \left\lfloor 0 \ \ 0 \right\rfloor[/itex]

The columns of [itex]V[/itex] are the eigenvectors of [itex]M^{\ T}M[/itex]. So I end up with the following:

[itex]V = \left\lceil -1 \ \ 1 \right\rceil[/itex]

[itex]\ \ \ \ \ \ \:\left\lfloor \ \ \:\: 1 \ \ 1 \right\rfloor[/itex]

The transpose of [itex]V[/itex] turns out to be no different than [itex]V[/itex]. To determine [itex]U[/itex], I find the eigenvectors of [itex]MM^{\ T}[/itex]. I end up with the following matrix:

[itex]U = \left\lceil 1 \ \ -1 \right\rceil[/itex]

[itex]\ \ \ \ \ \ \:\:\left\lfloor 1 \ \ \ \ \ \ 1 \right\rfloor[/itex]

When you scale the columns of [itex]U[/itex] and [itex]V^{\ T}[/itex] so that the matrices become orthogonal, you find that the product of [itex]U\Sigma V^{\ T}[/itex] multiplied by [itex]0.5[/itex]

*should*yield the matrix [itex]M[/itex]. It does not turn out to be [itex]M[/itex], though. Instead I end up with the following matrix:

[itex]0.5U\Sigma V^{\ T}= \left\lceil -1 \ \ 1 \right\rceil[/itex]

[itex]\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \:\,\,\,\left\lfloor -1 \ \ 1 \right\rfloor[/itex]

What am I doing wrong?!

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