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It is well known that eigenvalues for real symmetric matrices are the same for matrices $A$ and its transpose $A^\dagger$.

This made me wonder: Can I say the same about the singular values of a rectangular matrix? So basically, are the eigenvalues of $B B^\dagger$ the same as those of $B^\dagger B$?

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I know that $BB^T$ is not the transpose of $B^T B$. I was using the definition of the singular values of $B$ as the square-root of the eigenvalues of $B^T B$. Then, the singular values of $B^T$ should be $B B^T$, shouldn't they? – Lagerbaer Mar 31 '11 at 4:56
Oh, okay then. Sorry about that. – Arturo Magidin Mar 31 '11 at 4:58
up vote 10 down vote accepted

Both eigenvalues and singular values are invariant to matrix transpose no matter a matrix is square or rectangular.

The definition of eigenvalues of $A$ (must be square) is the $\lambda$ makes $$\det(\lambda I-A)=0$$ For $A^T$, $\det(\lambda I-A^T)=0$ is equivalent to $\det(\lambda I-A)=0$ since the determinant is invariant to matrix transpose. However, transpose does changes the eigenvectors.

It can also be demonstrated using Singular Value Decomposition. A matrix $A$ no matter square or rectangular can be decomposed as $$A=U\Sigma V^T$$ Its transpose can be decomposed as $A^T=V \Sigma^T U^T$. The transpose changes the singular vectors. But the singular values are persevered.

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Thanks. I should have looked at the SVD from the beginning. – Lagerbaer Mar 31 '11 at 4:58
Maybe it's a bit pedantic, but since it confused me for a moment maybe it's worth to point it out: if you define the singular values of $A$ as the square roots of the eigenvalues of $A^TA$ - as the OP does in the comment - then this is not true in general. A could have zero as singular value and $A^T$ not (or viceversa) probably define here singular values as square roots of the NON-ZERO eigenvalues of $A^TA$. – Hans Aug 20 '12 at 20:43
You brought up a good point. I didn't notice this before. I also would like to add a point. The most exact definition is this: non-zero singular values of $A$ are the square roots of the non-zero eigenvalues of $A^TA$ or $AA^T$. Note the singular value can also be zero. – Shiyu Aug 24 '12 at 3:24

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