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Results tagged with matrix-analysis
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user 17261
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
3
votes
Accepted
A simple but curious determinantal inequality
Let $C = A^{-(k+1)/2} B A B A^{-(k+1)/2}$ and $D = A^{-(k-1)/2} B A^{-1} B A^{-(k-1)/2}$ .
We have to show that $det(I+C) \ge det(I+D)$ .
Now my goal is to apply equation (5.21) in Ando, Majorizatio …
- 2,753
18
votes
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ ma...
Let $S$ be the span of $SO(N)$ .
Then it's obvious that if $A \in S$ and $D_1, D_2 \in SO(N)$ then $D_1^{-1} A D_2 \in S$ .
Therefore it's enough to show that $B := diag(1,0,...0) \in S$ .
If $N$ i …
- 2,753
4
votes
Accepted
A singular value-eigenvalue inequality
The conjecture is true.
Lemma 1 : For every matrix $Z$ holds $\lambda_j(Z^* Z + Z^* + Z ) \leq \lambda_j(Z^* Z + 2 (Z^* Z)^{1/2})$ .
For a proof see the proof of $\lambda_j(Z^* + Z ) \leq \lambda_j …
- 2,753
7
votes
Accepted
Well known matrix inequality?
The desired inequality follows from an majorisation argument :
According to Bhatia Matrix Analysis Corollary III.4.6 we have :
$log \lambda(AB) \succ log \lambda^\downarrow(A) + log \lambda^\uparro …
- 2,753