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$U\in \mathbb{C}^{n\times k}, k<n~$ and $~\lambda_i\in \mathbb{R} ~\forall~ i$

$U^*U=I_k$ but $UU^*$ is unknown.

Note that, $U$ is tall matrix formed by a few columns of some unitary matrix.

This matrix-form seems to be similar to an Eigen-decomposition, but I fail to see any relation between the eigenvalues of $~U^* diag(\lambda_1,\ldots,\lambda_n) U~$ and $(\lambda_1,\lambda_2,\ldots)$

Another observation (if it helps in anyway):

$U^*diag(\lambda_1,\ldots,\lambda_n)U$ also appears like a $k\times k$ sub-matrix of another $n\times n$ matrix, unitarily similar to $diag(\lambda_1,\ldots,\lambda_n)$.

In the worst case, I would want at least the maximum eigenvalue & eigenvector of it.

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  • $\begingroup$ Isn't it a singular value decomposition? $\endgroup$
    – Surb
    Commented Mar 18, 2015 at 9:44
  • $\begingroup$ Looks like so, but I need eigen values! $\endgroup$ Commented Mar 18, 2015 at 9:45
  • $\begingroup$ are the $\lambda_i$ complex or real? $\endgroup$
    – Surb
    Commented Mar 18, 2015 at 9:49
  • $\begingroup$ $\lambda_i$ are real values. $\endgroup$ Commented Mar 18, 2015 at 9:50
  • $\begingroup$ Note also that $U^*diag(\lambda_1,\ldots,\lambda_n)U$ is Hermitian and thus its singular values and eigenvalues corresponds (up to sign). $\endgroup$
    – Surb
    Commented Mar 18, 2015 at 9:58

1 Answer 1

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$\min\{\lambda_1,\ldots,\lambda_n\}\|x\|^2=\min\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2\leq x^*U^*diag(\lambda_1,\ldots,\lambda_n)Ux\leq \max\{\lambda_1,\ldots,\lambda_n\}\|Ux\|^2=\max\{\lambda_1,\ldots,\lambda_n\}\|x\|^2$

and therefore $\min\{\lambda_1,\ldots,\lambda_n\}\leq \lambda_i(S)\leq \max\{\lambda_1,\ldots,\lambda_n\}$ for all the eigenvalues of $S:=U^*diag(\lambda_1,\ldots,\lambda_n)U$.

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  • $\begingroup$ Is $V$ just invertible or is it unitary? If not unitary, eigenvalues of $V^*MV$ are not equal to that of $M$!! $\endgroup$ Commented Mar 18, 2015 at 11:40
  • $\begingroup$ You are right about my previous answer. I have edited it. $\endgroup$
    – RTJ
    Commented Mar 18, 2015 at 12:15

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