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I love mountainbike. I hate the Matrix Cookbook.


Apr
1
revised Solving $P=AB,Q=BA$, in the unknowns $A,B$
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Apr
1
asked Solving $P=AB,Q=BA$, in the unknowns $A,B$
Mar
10
comment Separating the eigenvalues of a Hermitian matrix with a special block structure
@Sam, can you show that $\det(J-xI)=\det((A-xI)(A-xI-\overline{B}(A-xI)^{-1}B))$ ?
Mar
10
comment Separating the eigenvalues of a Hermitian matrix with a special block structure
Sam, I think that you did not work really about your question. In particular, $J$ is "almost" a generic hermitian matrix and then, without supplementary hypothesis about $A,B$ (positive, definite ?), why this matrix $J$ would have a signature equal to $(n,n)$ under the hypothesis: $B$ is a real matrix ? It is obviously false: when $n=1$, the eigenvalues are $a\pm|b|$. Moreover, assume that $J$ has signature $(n,n)$ ; if $B$ is SLIGHTLY perturbed in the complex domain, then its signature is invariant. Observe seriously the case when $B$ is real.
Mar
7
comment Matrix equation XAX=B where the solution must be diagonal
Yes Suvrit, I agree. If $X$ is only assumed to be symmetric and $A,B$ are generic, then the minimum is $0$. That shows the interest to assume that $X$ is a diagonal matrix. Yet, I do not know if this hypothesis on $X$ is associated to (only) a mathematical exercise or to a practical use of Riccati equations.
Mar
6
comment Matrix equation XAX=B where the solution must be diagonal
@Suvrit, you write too quickly. Firstly, $X$ depends on $A,B$ ; in particular, if $A$ is invertible and $B$ is not, a solution $X$ of $XAX=B$ is certainly not generic. Secondly, if $A,B$ are generic in the PDS set, then $XAX=B$ has $2^n$ symmetric solutions and no diagonal solutions. Thirdly: read my edit.
Mar
6
revised Matrix equation XAX=B where the solution must be diagonal
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Mar
6
answered Matrix equation XAX=B where the solution must be diagonal
Mar
4
revised Mixing Numerical Range and inner product
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Mar
4
revised Mixing Numerical Range and inner product
added 393 characters in body
Mar
4
comment Mixing Numerical Range and inner product
The sequel proves nothing ! I give a method to solve your optimization problem: $\min_{u^Tu=1}u^TAU+b^Tu$. This method needs the calculation of $spectrum(A)$ and to find the roots of a polynomial of degree $2N-2$. Thus we have $2N-2$ candidates and we choose the one that realizes the minimum of $f$. Remark: your problem is not a convex one, even if $b=0$, because $\{u^Tu=1\}$ is not convex and $A$ is not necessarily $\geq 0$ or $\leq 0$.
Mar
3
answered Mixing Numerical Range and inner product
Feb
21
awarded  Critic
Feb
20
revised Regularized Gradient with respect to a matrix (with a specific structure)
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Feb
20
answered Regularized Gradient with respect to a matrix (with a specific structure)
Feb
8
comment Determine the expected size of a lower triangular sub-matrix of a random matrix?
Anthony, few remarks. i) For your first approximation, you assume $k/n$ is small. ii) The last inequality is $k<4\log(n)/|\log(p)|+1$. iii) About your "Since there is enough independence around": a choice $(I,J)$ means, in particular, a choice of the orderings of $I,J$. Then if you keep the sets $I,J$ and change only the orderings, you are far from independence, except again, if $k/n$ is very small, that is, if $p$ is small.
Jan
16
awarded  Yearling
Jan
12
comment LU decomposition
What is $\delta_{i,j}$ ?
Jan
11
comment LU decomposition
Even if your new conjecture, concerning a very specific case, is true, what is the interest ? I gave a method that is valid for any matrix and that has the same complexity than the deomposition LU. I am afraid that you are wasting your time.
Jan
10
revised LU decomposition
added 619 characters in body