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If you make an even-odd permutation, your matrix becomes $$ \begin{bmatrix} aI & I+Z\\ I+Z^{-1} & -aI \end{bmatrix}, $$ where $Z$ is the generator of the circulant algebra. Let $Z=FDF^{-1}$ be its eig …

If you are solving this numerically on a computer, you should use a variant of the Bartels-Stewart method; see for instance http://dl.acm.org/citation.cfm?id=146929 (Gardiner, Laub, Amato, Moler). Th …

(Probably this is not what you are asking for, but it might be a cue to modify your question in the right direction.) If you fix $A$, $\gamma$, $\phi$, then what you get is a linear system in the en …

The following Matlab code (unless I coded something wrong, which is well possible) finds a few random counterexamples each time I run it, even when restricted to off-diagonal entries: n = 3; for trie …

I don't think there is any computational advantage in general. For any pair of symmetric $M$, $N$ and given $c\neq d$, one can find symmetric $A, B$ such that $M=A+cB$, $N=A+dB$, so essentially you ar …

Let $A_2A_1^{-1}=U_A DV_A$ and $B_2B_1^{-1}=U_B DV_B$ be the two SVDs with the same $D$. Set $U_2'=U_AU_B'$, $U_1=V_B'V_A$, $V=B_1^{-1} U_1 A_1$. The first equality is clear. The second one is prove …

People in matrix analysis would call this a "canonical form under congruence". Take a look at http://arxiv.org/abs/0709.2473; the solution is stated there.

You do not need to show that the discs are disjoints. In fact, this won't hold for most diagonally dominant matrices, unlike the main result that you wish to prove. What you need is a stronger form o …

You might also be interested in checking out the representation of these operators with the Kronecker product and the vectorization map: $$ \operatorname{vec(AXB)}=(B^T \otimes A)\operatorname{vec}(X) …

You may think to the situation as a Gram-Schmidt orthonormalization partially performed (the columns of $U_1$ are already orthonormal), and complete it. The matrix $V=U_2-U_1U_1^TU_2$ has columns ort …

Why are you trying to avoid eigenvalue calculations in the first place? I think Arnoldi methods (such as Arpack, used e.g. in Matlab's eigs) would do a respectable job, and maybe even the power method …

Take a look at: Indefinite linear algebra and applications --- Israel Gohberg,Peter Lancaster,L. Rodman, section 4.2.

From Golub and Van Loan, Matrix Computations 3rd Edition, Section 12.5 ("updating the ULV decomposition": ...$O(n^3)$ flops are required to recompute the SVD of a matrix that has undergone a unit …

Converting previous comments into an answer, because I noticed they can fully answer the question. First of all, one can change basis to assume $A = [I\,\, 0]$, and partition $$ X = \begin{bmatrix}X_{ …

Not exactly my field, but I think that the same techniques used for Vandermonde's determinant work for Moore's and on finite rings (with the added complication of using FFTs over rings). A good refer …