4
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Invertibility of the Schur Complement
Yes, if you take the determinants, you obtain with
$$\operatorname{det}(M)=\operatorname{det}(M/ D)\cdot\operatorname{det}(D)
$$
therefore if $\operatorname{det}(M)$ is non-zero then $\operatorname{...
- 41
3
votes
Accepted
What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?
As for your first question, $S^{-1}$ is a diagonal block of $M^{-1}$. You can only say that its eigenvalues are interlaced with those of $M^{-1}$. Since all the eigenvalues of $M$ and $S$ are negative,...
- 52.3k
1
vote
Eigenvalues of a block matrix composed of Toeplitz matrices
Introduce the projection operator $P$ such that $A=PMP$, $Q=1-P$, and $C=QMQ$. Next consider the self-energy operator $\Sigma_P(E)=PMQ(E-QMQ)^{-1}QMP$.The map $M\rightarrow PMP+\Sigma_P(E)$ relates ...
- 492
1
vote
Transform a matrix optimization problem into a semidefinite programming
$\newcommand\diag{\operatorname{diag}}$Write $X^{-1}$ as $Y_1-Y_2$, where $Y_1$ and $Y_2$ are positive-semidefinite symmetric matrices.
Detail: This can be done for any nonsingular symmetric matrix $...
- 128k
1
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Is it impossible for determinants of these matrices to both be negative?
OK, second try, equation references are pointing to my other answer:
Using the definitions (3) for $\alpha_j$ and $\beta_j$, we can proof the OPs conjecture in the following way: As the determinants
$$...
- 3,691
1
vote
Accepted
Is it impossible for determinants of these matrices to both be negative?
This is a partial answer, now with added material.
Assume that $n$ is a multiple of 3, the other two cases should be similar. I'll denote $C_1\mapsto a$ and $C_2\mapsto b$, such that for, e.g., $n=6$,
...
- 3,691
1
vote
Accepted
Two-level correlation function of eigenvalues for large random matrices
Let me address the issue raised by the OP of the universality of the two-point correlation function.
The universality of $\rho^{(2)}(\lambda,\mu)$ does exist if one considers the correlations locally, ...
- 188k
1
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Sufficient conditions for invertibility of a block tridiagonal matrix
The following is a list of answers I know for some specific cases. However, they are not strong enough for my uses.
Simple conditions
A sufficient, but weak condition is that $C_i = 0$ for each $i$.
...
- 397
1
vote
Accepted
Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$
As the comments note, what you are doing is Gaussian elimination with complete pivoting: permute rows and column to bring an 1 to the top-left corner, make one step of Gaussian elimination, repeat.
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- 20.2k
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