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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Two State Implicit Filter Derivation

I've been reading The "Two-State Implicit Filter – Recursive Estimation for Mobile Robots" by Bloesch. The gist of the paper is it tries to find the balance between a Kalman Filter and a ...
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Matrix involving reciprocal factorials

Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$. Is there a formula for the ...
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Minimizing symmetric convex functions of eigenvalues

I am stuck with the following problem. Prove that the optimal value to the SDP \begin{align} \text{minimize} \quad &\operatorname{tr}(V) \end{align} \begin{align} \text{subject to} \quad &\...
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Converting nonlinear matrix inequality into an LMI

I need the following inequality to hold; $$ M+(CK)^{\text{T}}NCK > 0 $$ where $$ C \in \mathbb{R}^{m \times p}, ~~~ N=N^{\text{T}} > 0 \in \mathbb{R}^{m \times m} $$ are known, and $$ M=M^{\text{...
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Schur complement, Congruence Transformation and Positive Semi definiteness

The Schur Complement of a block matrix $A$ in $X$ where $$ X = \begin{bmatrix} A & B \\ B^T& C \end{bmatrix} $$ is defined as $S = C-B^TA^{-1}B$ whenever $A$ is non-singular. ...
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Single constraint quadratic optimization dual form expression using the Schur complement

Strong duality result for non-convex problem with two quadratic functions is a related question. However, I am trying to understand how the dual form problem comes about. This dual form ...
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Recover a matrix from its Schur complements

Suppose I have a matrix: $$ M = \begin{bmatrix}A & B \\ C & D\end{bmatrix} $$ With Schur complements: $$ M/A = D - CA^{-1}B \\ M/D = A - BD^{-1}C \\ $$ Given only the Schur complements $M/A$ ...
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Is this a known operation on matrices?

In my work on Gaussian processes, I encountered the following operation on matrices. Let $K$ be a positive definite $d \times d$ matrix. Define $$ \tilde{K}_{ij} = K_{ij} - \frac{(K_{ki} + K_{li}) (K_{...
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How can we ensure the matrix $\mathbf{T} - ( \mathbf{C} \circ \mathbf{A} )( \mathbf{C} \circ \mathbf{A} )^H$ is positive semi-definite(PSD)

Specifically, the operator $\circ$ denotes the hadamard product, the matrix $\mathbf{T}$ is a low rank toeplitz PSD matrix , the matrix $\mathbf{A}$ is a matrix in vandermonde structure and its ...
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Schur complement for negative semidefinite

I have: $$X^{-1}(1-\alpha^2) - A^T(I+AXA^T)^{-1}A\preceq0,$$ where $X\succ0$, $\alpha$ is scalar, and $A$ is arbitrary matrix. Can I find an equivalent condition for the inequlity using the Schur ...
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For ($n \times p$) $A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \\ \end{bmatrix} $, show that $A_{22} = A_{21}A_{11}^{-1}A_{12}$

I'm currently trying to solve the following problem. $A$ is $n \times p$ matrix with rank $r < \text{min}(n,p)$ and $A$ is partitioned as follows. $$A = \begin{bmatrix} A_{11} & A_{12} \\ A_{...
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Using Schur's Complement to Reformulate Semidefinite Programming Constraint

I am currently reading Boyd & Vandenberghe's Convex Optimization and encountered (at least twice) reformulations of matrix inequalities that confuses me a lot. I will try to clarify my question ...
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Is there a Schur-like theorem for proving $B^TB\succeq A$?

This is in the context of semidefinite programs, and $A\in\mathbb R^{n\times n},~B\in\mathbb R^{k\times n}$ are variables. If want to show that $A-B^TB\succeq 0$ and I know $A\succeq 0$, then I can ...
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Proving the equation of positive definite submatrix

For $n \ge 2$, $A$ and $A^{-1}$ are $n \times n$ positive definite matrices. Now, for a scalar $\alpha > 0$ and $(n-1)$-dimensional column vector $\beta$ and $(n-1) \times (n-1)$ $\Delta$ matrix. ...
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Positive semidefinite block matrix

Let $M$ be an $n \times n$ block matrix defined as $$M = \begin{bmatrix} A & \mathbf{b} \\ \mathbf{b}^T & 1 \end{bmatrix}$$ where $A$ is an invertible and symmetric $(n-1) \times (n-1)$ matrix,...
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On the relationship between the Sylvester criterion and the Schur complement for positive semi-definiteness of a matrix via LMI

For some context, I see that when people try to determine the positive definiteness of a matrix, the Sylvester criterion can be of great help since the determinant is easy to calculate. For the case ...
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A characterization of symmetric positive definite matrices using Schur complements [closed]

I have the matrix $$X = \begin{pmatrix} A & B\\ B^\intercal & C\end{pmatrix}\in \mathbb R^{nxn}$$ and $S:= C-B^\intercal A^{-1}B$. by considering $$\min_u \quad u^\intercal Au + 2v^\intercal B^...
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Involution on $2\times 2$ matrices

Show that the map on $2\times 2$ matrices \begin{eqnarray} \left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
orangeskid's user avatar
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Schur complement of the marginalized normal covariance matrix given joint Cholesky decomposition

Consider a multivariate normal distribution with covariance matrix $\Sigma$ of size $n \times n$, which can be written in terms of its lower triangular Cholesky decomposition $L$ as $$\Sigma = L \cdot ...
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Relationship of determinants of block matrices

I am thinking about the following problem: Let $$M = \begin{bmatrix}A & X_{12} & X_{13} \\ X_{21} & B & X_{23} \\ X_{31} & X_{32} & C\end{bmatrix} \in \mathbb{R}_{\geq0}^{n \...
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Determinant of product of matrix and nullspace

Assume I have a symmetric, positive-definite matrix $S \in \mathbb{R}^{p \times p}$. Assume that there is some matrix $L \in \mathbb{R}^{n \times p}$ that has full row-rank, i.e., has rank $n$ and ...
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Schur complement

I am trying to understand what steps need to be done to go from $P-A^TPA\succ0$ (with $P \succ 0$ and $G$ can be any matrix) to $$\begin{bmatrix} P & A^TG^T \\ GA& G + G^T - P \end{bmatrix} \...
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Determinant of block matrices.

$$ X = \begin{pmatrix} 1+b_1 & 1 & 0 & 0 & 0 & \frac{1}{a_{6}} \\ 1+b_2 & 1 & 1 & 0 & 0 & -\frac{a_1}{a_6} \\ b_3 & 1 & 1 & 1 & 0 & -\frac{...
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How to prove the simplified form of $det\left( \begin{matrix} A& \vec{x}\\ \vec{y}^T& a\\ \end{matrix} \right) $?

The question is, how to prove $det\left( \begin{matrix} A& \vec{x}\\ \vec{y}^T& a\\ \end{matrix} \right) =a\cdot det\left( A \right) -\vec{y}^T\left( adj\left( A \right) \right) \vec{x}$, ...
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LMI reformulation

In Data-driven stabilization of discrete-time control-affine nonlinear systems: a Koopman operator approach, I read that the following LMI $$\left(\begin{array}{cccc} \mathbb{U}^{\top} P \mathbb{U}-P &...
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Matrix identity involving inverse

Let matrix $Y \in \mathbb{S}^{n}$ be symmetric and positive definite (thus invertible) and $X \in \mathbb{R}^{n \times n}$ such that $Y - X X^T \succ 0$. I think that the following identity should ...
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Schur complement and positive semidefinite cones

I know that my question might be trivial but I would appreciate your feedback. I know that the Schur complement can be used to express a quadratic inequality as a positive semidefinite matrix and vice-...
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Applying the inverse of the Schur complement without matrix-matrix products

If one attempts to solve a (block matrix) saddle point problem such as $$\begin{bmatrix} A & -B^T \\ B & 0 \end{bmatrix} \begin{bmatrix} u \\ p \end{bmatrix} = \begin{bmatrix} f \\ 0 \end{...
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Does $(I-BB^\dagger)C=0$ hold for a symmetric positive semidefinite matrix $G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$ with $C \preceq B$?

Given a symmetric positive semidefinite matrix $$ G = \begin{pmatrix} A & B \\ B^T &C \end{pmatrix}$$ where $A$, $B$ and $C$ are not invertible, and $C\preceq B$, does the following equality ...
Niz's user avatar
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How to prove that a certain block matrix is positive semi definite, which depends on a undetermined submatrix

How should I proof the following matrix $$M = \begin{pmatrix} Z-A^TZA & -A^TZB\\ -B^TZA & -B^TZB \end{pmatrix},$$ to be positive semidefinite? The matrices $A\in \mathbb{R}^{n\times n}$ and $ ...
HerChip's user avatar
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Low Rank Update of the Schur Complement

Suppose we have an $m \times n$ matrix $A$ represented in block form as $$ A = \left[\begin{array}{cc} A_{11} & A_{12} \\ A_{21} & A_{22} \end{array}\right]$$ where $$A_{11}$$ is $k\times k$. ...
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Schur complement like operation on a singular matrix

For the classical definition of matrix inversion by Schur complement, given by: \begin{aligned} M^{-1}=\left[\begin{array}{ll} A & B \\ C & D \end{array}\right]^{-1} &=\left(\left[\begin{...
Audrey's user avatar
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Using a Schur complement, prove that the matrix has two double eigenvalues

For a skew symmetric block $n \times n$ matrix $B$, prove that matrix $M$ has two double eigenvalues. $$ M = \begin{bmatrix} I & B \\ B & I\end{bmatrix} $$ For a proof, I was using the ...
Engineeringbridges's user avatar
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Failure to invert sparse matrix

I have a large block arrowhead matrix which has significant sparsity in the following pattern: $\mathbf{M} = \left( \begin{array}{c|c} \mathbf{A} & \mathbf{B}^{\top}\\ \hline \mathbf{B} & \...
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Help with SDP and Schur's Complement

I'm trying so hard to understand SDP and how Schur's complement is used and what does it even mean? Is there a good and simple reference with some numerical examples that can answer my question ...
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Schur complement for linear matrix inequality (LMI)

Given the following inequality \begin{align} & \gamma \left( Q - (A Q + BY)^T Q^{-1} (A Q + BY) \right) - Y^T R^{1 \over 2} R^{1\over 2} Y - Q Q_1^{1\over2} Q_1^{1\over2} Q \succeq 0 \tag{1} \end{...
Xero Smith's user avatar
3 votes
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Lower-bounding minimal eigenvalue via the Schur complement

Suppose that $$M=\left( \begin{array}{cc} A & B\\ B^\top & C \end{array} \right)$$ for some symmetric matrices $A$ and $C$, and $C$ is invertible. Is it true that: $$\lambda_{\min}(M) \ge \...
Probabilist's user avatar
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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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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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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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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_{...
Moha's user avatar
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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 ...
Moe's user avatar
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The Schur complement is matrix concave

I am confused regarding the Schur complement. From Boyd and Vandenberghe's Convex Optimization, Suppose $X \in S^n_{++}$ partitioned as $$X = \begin{bmatrix} A & B\\ B^T & C\\ \end{bmatrix}$$ ...
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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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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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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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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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2 answers
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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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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 \...
Kumar's user avatar
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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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