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Given the eigendecompositions $AA^{\top}=Q \Lambda Q^{\top}$ and $A^{\top}A=P \Lambda P^{\top}$, where $\Lambda$ is a diagonal matrix (of eigenvalues) and $P$ and $Q$ are unitary eigenvectors matrices of $A^{\top}A$ and $AA^{\top}$, is there a nice way to show that $A^{\top}Q \Lambda^{-1} Q^{\top}=P \Lambda^{-1} P^{\top}A^{\top}$ ?


I originally obtained this by solving the following two optimization problems:

Define:

$A$ is an $m$ by $n$ matrix, $x$ a $n$ by 1 vector, and $y$,$\hat y$ are $m$ by $1$ vectors.

Assume the following two optimization problems:

a) Minimize $x^{\top}x$ given the constraint that $Ax=y$

b) Minimize $(y-\hat y)^{\top}(y-\hat y)$ where $\hat y=Ax$.

The solution to both of these problems can be shown to be $x_{opt}=Wy$ where $W$ is the Moore–Penrose pseudoinverse of matrix $A$.

what is a good reference (a textbook or published paper) that discusses the above problems ?

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    See http://math.stackexchange.com/questions/49857/pseudo-inverse-solution-for-linear-equation-system-using-the-svd2011-12-17

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