# 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.

127 questions
Filter by
Sorted by
Tagged with
37 views

### 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 ...
• 276
48 views

### 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 ...
• 644
1 vote
67 views

### 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 &\...
• 465
59 views

• 258
15 views

### 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 ...
53 views

### 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 ...
• 616
120 views

• 19
78 views

### 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{...
• 54k
1 vote
143 views

• 83
1 vote
48 views

### 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 ...
72 views

• 4,972
53 views

### 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 $det\left( \begin{matrix} A& \vec{x}\\ \vec{y}^T& a\\ \end{matrix} \right) =a\cdot det\left( A \right) -\vec{y}^T\left( adj\left( A \right) \right) \vec{x}$, ...
• 211
67 views

• 523
81 views

### 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 ...
• 87
99 views

• 95
440 views

### 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 ...
1 vote
1k views

### 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{...
• 111
613 views

• 100
1 vote
119 views

• 33
150 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 ...
• 35
690 views

### 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 $$X = \begin{bmatrix} A & B\\ B^T & C\\ \end{bmatrix}$$ ...
• 506
110 views

### 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 ...
• 616
1 vote
73 views

### 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: ...
• 469
1 vote
72 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 $p$ ...
• 663
1 vote
$$\begin{array}{ll} \text{minimize} & \dfrac{(c^T x)^2}{(d^Tx)}\\ \text{subject to} & Ax \leq b\\ & d^T x > 0\end{array}$$ I have been stuck on this question for a couple days. I am ...