Questions tagged [schur-complement]

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1answer
91 views

What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?

I have two problems related to eigenvalues of negative definite matrices: I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
1
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0answers
82 views

Positive definiteness of a Matrix

$K>0\in \mathbb{R}^{n\times n}$, $P>0 \in \mathbb{R}^{n\times n}$ are diagonal positive definite matrices. And $R\geq 0\in \mathbb{R}^{m\times m}$ is positive semi-definite matrix. Let $B\in \...
2
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1answer
347 views

Eigenvalues of a block matrix composed of Toeplitz matrices

If I have a block matrix of the form $$ M = \begin{pmatrix} A &B \\[6pt] -B & C \end{pmatrix} $$ and if $A$ is invertible I can write determinant in terms of the Schur ...
5
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1answer
341 views

Invertibility of the Schur Complement

Suppose that $$ M = \begin{bmatrix}A & B \\ C & D\end{bmatrix}. $$ I know that if $D$ and $M\setminus D$ (where $M\setminus D$ is the Schur Complement of $D$ in $M$) are invertible, then $M$...
4
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0answers
398 views

Determining whether a Schur complement is invertible

Consider the symmetric matrix $$M = \begin{bmatrix} A & B \\ B^T & -C \end{bmatrix}$$ where $A \in \cal{R}^{n \times n}$ and $C \in \cal{R}^{m\times m}$ are symmetric, ...
1
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0answers
438 views

Why are SDP generally slow?

This is more of a conceptual question. Don't expect a highly mathematical question. Nonetheless, the questions I pose here often arise in my field (not mathematics). Usually Semidefinite Programs (...
4
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0answers
65 views

successive schur complements

If I have a large (e.g. 6000x6000), sparse, positive definite matrix $M$ (which may have individual entries everywhere, but most non-zero entries are on / around the diagional). Divide $M$ into blocks ...
0
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1answer
658 views

Bounding a determinant ratio

Let $A=[A_{0}\ E;E^{T} \ B]$ be a real positive definite matrix and let $B$ be a principal submatrix. I am interested in tightly bounding $\frac{|B|}{|A|}$ from below in some "explicit" way that will ...
2
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1answer
573 views

When is a Schur complement an $M$-matrix?

Let $F=\begin{bmatrix}A & B \\\\ B^{T} & D\end{bmatrix}$ be symmetric and strictly diagonally dominant (thus an $H$-matrix). I also know that $B>0$ entrywise. What I am trying to show is ...