# Questions tagged [schur-complement]

The Schur-complement captures several relationships between the properties of a block matrix and the properties of its blocks, such as semi-definiteness, Cholesky decomposition, etc.

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### A necessary and sufficient condition for a symmetric matrix to be positive semidefinite in terms of its Schur complement

According to the Wikipedia article Schur complement, if $X$ is a symmetric matrix of real numbers given by $X = \begin{bmatrix} A &B\\ B^T &C \end{bmatrix}$, then $X \succeq0$ (i.e. $X$ is ...
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### What is Schur complement? [closed]

I have the following query about Schur complement. Could anyone explain the following problem to me? This picture is from the following paper. --> https://www.mdpi.com/2076-3417/10/12/4270 Schur ...
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### How to prove that the Schur complement of symmetric, positive-definite matrix is positive-definite?

We have $A \in \mathbb{R}^{n \times n}$ which is symmetric and positive-definite. Also, $A$ is a block matrix: $$A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{pmatrix}$$ I ...
1answer
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### How to compute the determinant of this block matrix?

$$M = \left[\begin{matrix} -C & -A \\ A^\top & 0 \end{matrix} \right]$$ I found a paper using $\det(M) = \det(A^\top C^{-1}A)$ but don't know how to prove this.
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### Maximum upper left sub-matrix with zeros by row and column permutation

I have a square matrix filled with zeros and ones and I am allowed to permute the row and column order with the same permutation for rows and columns. The goal is to find a permutation such that there ...
3answers
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### Discrete-time LQR and solutions via LMI

Having a infinite horizon discrete-time LQR problem $J^* = \min_u \ \sum_{k=0}^{\infty} x^\top_k Q x_k +u^\top_kRu_k$ subject to $x_{k+1}= Ax_k+Bu_k, \quad x(0)=x_0$. With some algebra ...
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### Schur complement in LUP decomposition of a block tridiagonal matrix

Section 2.2 of the article On twisted factorizations of block tridiagonal matrices explains how to do a LUP decomposition of block tridiagonal matrices by showing the process on a 4 blocks by 4 blocks ...
2answers
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### Connectedness and invertibility of Laplacian matrix

Context: In the context of circuit theory and graph theory, suppose we have a graph $G,$ then the Laplacian (Kirchhoff) matrix $L$ is defined as follows: $$L = D-A \tag{1}$$ where $D$ is the ...
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### How to write this inequality in terms of Schur Complement?

I know the basis about Schur-Complement. Anyway, while looking at this inequality to apply it in order to solve for $\lambda$ such that the the matrix is definite positive, I got a little bit confused ...
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### Can $A + uv^T$ be non-singular if A(order n) has rank n-1

Suppose $A$ is a matrix of order $n$ with rank n-1, $u$ and $v$ are $n$-vectors. Can $A + uv^T$ be nonsingular, if yes then find such $u$ and $v$. First I wrote Rank$(A + uv^T) <=$ Rank$(A)$ + ...
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### Schur complement condition number

How can I most simply show, without referring to advanced theorems, that the Schur complement is better-conditioned than the original SPD matrix itself?
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