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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Help in understanding the use of Schur's complement and the results

I would love if you can help me understand the following result I found in article I read (https://www.jmlr.org/papers/volume6/chechik05a/chechik05a.pdf).(All variables are multivariate gaussian) $\ ...
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55 views

Partitioned positive definite matrix property

I am interested in the following problem: Let $X$ be a real symmetric positive definite matrix partitioned into four submatrices as follows: $$ X = \begin{pmatrix} A&B\\ B^T & C \...
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34 views

Reformulation of LMI

In a paper I have read, the authors reformulated the following LMI $$X_{t} \succeq \left[\begin{array}{cc}\alpha_{i}\left(B_{t} U+C_{t}\right)^{\top} E_{i}\left(B_{t} U+C_{t}\right) & \left( B_{t} ...
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48 views

Optimisation of nonlinear matrix objective with semidefinite constraints (classical Fisher information)

Given the invertible symmetric block diagonal matrices $S = S_1\oplus S_2$ and $dS = dS_1 \oplus dS_2$, I want to find the symmetric matrix X with blocks $$ X = \left[ \begin{array}[cc] ~X_1 & X_{...
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36 views

Inversibility of a block matrix in schur decomposition

Let $A'$ be a given, n × n, real, positive definite matrix partitioned as follows: \begin{pmatrix} A & B \\ B^T & C \end{pmatrix} show that $C − B^TA^{−1}B$ is positive definite. I know that I ...
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92 views

Schur-complement: matrix convex vs matrix concave

I have some confusion regarding schur complement. In the book, Convex Optimization by Boyd and Vandenberghe, there is a question as follows: Suppose $X \in S^n_{++}$ partitioned as $X= \Bigg[\begin{...
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Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?

I have the constraint \begin{align} X - Y A^\dagger Y^T\succ0, \end{align} where $A^\dagger$ is the pseudoinverse of $A\succeq0$. Can we still use the Schur complement to write the constraint as an ...
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Estimation of capacitance coefficient — Woodbury identity

I would like to find a non-trivial lower bound on the following quantity: \begin{align} \sigma^2 - x_i^T M^{-1}x_i, \tag{1} \end{align} By assumption $ \sigma^2 - x_i^T M^{-1}x_i \geq 0$, but I am ...
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Upper bounding the largest singular value of a matrix $X$ via LMI — is it correct?

For $z \in \Bbb C$ and $\delta > 0$, the inequality $|z| < \delta$ is equivalent to the matrix inequality $$\begin{bmatrix} -\delta & z\\ z^* & -\delta \end{bmatrix} \prec 0$$ (Source: ...
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60 views

Finding $\text{diag} \lbrace \textbf{s} \rbrace$ that makes block matrix positive definite

Let $$ X =\begin{pmatrix} \Sigma & \Sigma - \text{diag} \lbrace \textbf{s} \rbrace \\ \Sigma - \text{diag} \lbrace \textbf{s} \rbrace & \Sigma \end{pmatrix} $$ where $\textbf{s}$ is a $p$ ...
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76 views

How do I transform this problem into a semidefinite program?

$$\begin{array}{ll} \text{minimize} & \dfrac{(c^T x)^2}{(d^Tx)}\\ \text{subject to} & Ax \leq b\\ & d^T x > 0\end{array}$$ I have been stuck on this question for a couple days. I am ...
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145 views

Positive eigenvalues and Schur complements

For a symmetric matrix, $$M = \left(\begin{array}{cc} A & C\\ C^{\top} & D \end{array}\right)$$ it is well known that $M$ is positive definite if and only if $A$ and the Shur complement $...
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Schur complement question.

Good evening. Before start, I would like to pray for the end of the COVID19 virus soon. I have a question related to the calculation of one Schur complement matrix. Here is the matrix I'm interested ...
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Transformations in LMI constraints

Suppose that $T$ is a symmetric and invertible matrix. The following must be true \begin{equation}\label{eq:SDP}\left[\begin{array}{cc} X & A \\ A^{\top} & B \end{array}\right] \succeq 0 \...
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79 views

A sufficient condition for a block diagonal matrix to be positive definite

Let $$P = \begin{bmatrix} A & B \\ B^\top & D\end{bmatrix}$$ where blocks $A$ and $D$ are positive definite. All the matrices $A$, $B$ and $D$ are $n \times n$. I was wondering if one could ...
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190 views

About Schur theorem

Schur theorem: Let $T$ be a linear operator on a finite-dimensional inner product space $V$, Suppose that the characteristic polynomial of $T$ splits, then there exists an orthonormal basis $\beta$ ...
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108 views

Converting a quadratically constrained optimization problem into a standard semidefinite program

I have a constrained matrix optimization problem as follows \begin{align} \max\limits_{X,Y} \;\; &tr\Big( X^T B X \Lambda \Big) + tr\Big( BY\Big) + tr\Big( X^T C \Lambda \Big) \\ \text{subject ...
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How to prove the $n\times n$ matrix $A=\big(\frac{1}{i+j+1}\big)_{i,j\in [n]}$ is positive semi-definite? [duplicate]

We've been trying to show that the matrix $$A=\Big(\frac{1}{i+j+1}\Big)_{i,j\in [n]}$$ is positive semi-definite. We've tried induction on $n$ using the Schur complement, but there is no simple ...
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Schur Complement and Positive Semidefiniteness

Suppose $\mathbf{D} = \begin{bmatrix} \textbf{A} & \textbf{b} \\ \textbf{b}^T & c \end{bmatrix}$ where $\textbf{A} \in \mathbb{R}^{n \times n}$, $\textbf{b} \in \mathbb{R}^n$ and $c \in \...
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Relationship between conditional probability and conditional with complements

Does P(A|B^c) = 1 - P(A|B)??? Where B^c denotes B complement. Can someone show work to show why it is true or not true? I intuitively think they are not equal, but I'd like some confirmation.
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Variant of the Schur Complement

Let $$M = \left(\begin{array}{cc}{A} & {B} \\ {B^{T}} & {C}\end{array}\right)$$ be symmetric, and let $A$ be invertible. Then the Schur Complement Lemma suggests that $$C-B^{T} A^{-1} B \...
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73 views

Schur complements for nonstrict inequalities

I am trying to understand the following proof from the book "Linear Matrix Inequalities in System and Control Theory". However I am struggling to understand why $S_{2}$ must equal zero. Why isn'...
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697 views

Eigenvalues of the Schur complement

For a symmetric invertible block matrix such as below, is there a relation between the eigenvalues of $M$ and that of the Schur complements and the matrices in the diagonal? \begin{align} M = \begin{...
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100 views

Positive semidefiniteness of block matrix when diagonal blocks are not invertible

Let $$M =\left[\begin{array}{cc} A & B\\ B^{T} & D\end{array}\right]$$ where blocks $A$ and $D$ are not invertible, but both are positive semidefinite. Are there conditions such that $M$ is ...
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68 views

$C= A\oplus B$. Then prove that $C$ is diagonalisable if and only if both $ A$ and $B$ are diagonalisable.

Let $A\in M_n(R)$ and $B\in M_m(R)$. Suppose $C= A\oplus B$. Then prove that $C$ is diagonalisable if and only if both $ A$ and $B$ are diagonalisable. Let $C$ be diagonalisable, so $\exists P=...
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728 views

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 ...
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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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If a matrix has eigenvalues with non-zero real parts, can the eigenvalues of its Schur complement be arbitrarily close to zero?

Given symmetric $A\in\mathbb{R}^{m\times m}$, $B\in\mathbb{R}^{m\times n}$, symmetric $D\in\mathbb{R}^{n\times n}$, assume the eigenvalues $\lambda_i\in [\alpha_4,\alpha_3]\times[-\beta_2,\beta_2]\...
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Does a certain matrix inequality imply that a matrix is negative definite?

Suppose I have the matrix \begin{align} F_s&=\begin{bmatrix} F_{1s} & G \\ G^T & F_{2s} \end{bmatrix} \end{align} where $F_{1s},F_{2s}$ are symmetric matrices that are also negative ...
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Formulating a convex optimization problem as semidefinite program

I have the following minimization problem $$\text{minimize} \quad f(x)= c^T F(x)^{-1} c$$ where $F : \mathbb R^n \to \mbox{Sym}_m (\mathbb R)$, $$\mbox{dom} f = \{x \in \mathbb{R}^n \mid F(x) \succ ...
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How can we write this inequality as an LMI?

Let $X$, $Y$ and $Z$ be positive definite matrices. How can we write the following inequality as an LMI? $$XY - Z^2 - I \succ 0$$ Here, $I$ is the identity matrix. For example, if it was $XY-Z^2Y-I&...
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how to subtract -13-4 using complements method?

i came up to these question on my homework: Complete needed actions for these decimal numbers using complementary arithmetic, to get the right result. ...
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$\forall C\in \mathbb{R}^{n\times n}, \ \ \ PAP-CMC^T\geq 0\ \ \ \Longrightarrow\ \ \ A-CMC^T\geq 0$ [closed]

let M be a positive definite $n\times n$ matrix and A a positive semidefinite $n\times n$ matrix and P is an orthogonal projector of some subspace of $\mathbb{R}^n$ into $\mathbb{R}^n$ so is this ...
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Standard notation for $X-E[X|Y]$

Is there standard notation for a random variable of the form $X-E[X|Y]$?
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About the use of Schur's complement. Why they are equivalent?

$$y^{\top}Qy+y^{\top}q+r\geq -ay^{\top}x-b, \qquad \forall y \in \mathbb R^{n}$$ where $Q \succeq 0$. One can use Schur's complement to replace it by an equivalent linear matrix inequality (LMI). $$...
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183 views

using Schur's complement and Young's inequality to reduce matrix algebraic equation to LMI

This questions concerns the practical implementation of schur's complement and Young's inequality. Consider the following \begin{align} \begin{pmatrix} \begin{pmatrix} \mathbb{A}_{\mathbb{Z}} & ...
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Finite horizon Riccati solved by LMI

Considering the system $x_{k+1}=Ax_k+Bu_k$ with quadratic cost $J^* = \min x_N^T S x_N + \sum_{k=0}^{N-1} x_k^T Qx_k+u^T_kRu_k$ where $Q,S\succeq 0, R\succ 0$. The optimal state feedback is found ...
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Elements of positive definite matrix

Consider the symmetric matrix \begin{equation} \begin{bmatrix} a & b\\ b^T & c \end{bmatrix}\succ 0 \end{equation} By assumption $c\succ 0$. I want to prove that $a\succ 0$. For the Schur ...
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Schur complement without inverted term and YALMIP solving

first of all, excuse my poor English :( Anyway, I need to factorize the equation below in terms of Schur Complement. $\begin{equation} { x }^{ T }\left( PA+{ A }^{ T }P \right) x<-\gamma { x }^{ ...
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303 views

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 ...
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578 views

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 ...
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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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120 views

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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141 views

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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154 views

$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$

We know that the conditional variance of a multivariate normal vector $(X,Y)$ is equal to the Schur complement: $$V(X|Y)=\Sigma_{XX}-\Sigma_{XY}\Sigma_{YY}^{-1}\Sigma_{YX}$$ However, $\Sigma_{XX}-\...
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49 views

the inverse of a sum of two symmetric for schur completion?

I have a up-triangulate Jacobi matrix J which can be blocked like : $J = \begin{bmatrix}A & B\\ 0 & C\end{bmatrix} $ both A and C are up-triangulate, we can get Hessian matrix H by: $H = J'...