Questions tagged [schur-complement]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
6 votes
2 answers
930 views

Is it impossible for determinants of these matrices to both be negative?

Suppose $A,B \in M_{n}(\Bbb{R})$ such that $A = \left[C_{1}\middle|\frac{I}{0\dots0}\right], B= \left[C_{2}\middle|\frac{I}{0\dots0}\right]$ , where $A$ and $B$ have different first columns (...
BAYMAX's user avatar
  • 51
2 votes
1 answer
276 views

Two-level correlation function of eigenvalues for large random matrices

One can define the density of eigenvalues of a $N\times N$ Hermitian random matrix $H$ as: \begin{equation} \rho(\lambda)=\left \langle\frac{1}{N} \operatorname{Tr} \delta(\lambda-H)\right\rangle \end{...
Matt's user avatar
  • 97
4 votes
2 answers
857 views

Sufficient conditions for invertibility of a block tridiagonal matrix

Let $M_n \in \mathbb{R}^{N \times N}$ be a block-tridiagonal matrix: $$M_n = \begin{bmatrix} B_1 & C_1 & 0 & 0 & \cdots & 0 \\ A_1 & B_2 & C_2 & 0 & \cdots & 0 \...
kaba's user avatar
  • 387
1 vote
1 answer
140 views

Schur complement and depermuting an algorithm for $\mathsf{determinant}\bmod2$

Let $$M=\begin{bmatrix}A&B\\C&D\end{bmatrix}$$ be a matrix in $\mathbb F_2^{n\times n}$ where $A\in\mathbb F_2$ and $D\in\mathbb F_2^{(n-1)\times(n-1)}$ are square. Through the determinant ...
Turbo's user avatar
  • 13.6k
3 votes
1 answer
514 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=[...
Parikshit's user avatar
1 vote
0 answers
120 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 \...
kosa's user avatar
  • 111
3 votes
1 answer
618 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 ...
zylatis's user avatar
  • 141
6 votes
1 answer
1k 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 $M$...
Nathan's user avatar
  • 61
4 votes
0 answers
556 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, ...
berkin's user avatar
  • 41
1 vote
0 answers
607 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 (...
mermeladeK's user avatar
4 votes
0 answers
90 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 ...
Jakob's user avatar
  • 41
0 votes
1 answer
836 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 ...
Felix Goldberg's user avatar
2 votes
1 answer
675 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 ...
Felix Goldberg's user avatar