# Questions tagged [schur-complement]

The schur-complement tag has no usage guidance.

9
questions

**2**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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 ...