Mathematics

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.

132 questions
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Inverse of a $2 \times 2$ block matrix

Let $$S := \pmatrix{A&B\\C&D}$$ If $A^{-1}$ or $D^{-1}$ exist, we know that matrix $S$ can be inverted. $$S^{-1} = \pmatrix{A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1}&-A^{-1}B(D-CA^{-1}B)^{-1}\\-(D-CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}}$$ But, what if…
Claire
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A geometric way to reason about Schur complements?

I am trying to understand some theorems whose standard proofs seem to involve extensive matrix manipulations. I'm finding it a bit difficult to see a big picture behind all the matrix algebra, and I'd like to ask two questions which will hopefully…
robinson
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Determinant of block matrix with singular blocks on the diagonal

Let $A$ and $D$ be square matrices, and let $B$ and $C$ be matrices of valid shapes to allow the formation of $$ M = \begin{bmatrix} A & B \\ C & D \end{bmatrix}. $$ If $\det{A}\neq0$, we may use the Schur complement to express $\det{M}$ in terms of…
Mårten W
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Efficiently computing Schur complement

I would like to compute the Schur complement $A-B^TC^{-1}B$ , where $C = I+VV^T$ (diagonal plus low rank). The matrix $A$ has $10^3$ rows/columns, while $C$ has $10^6$ rows/columns. The Woodbury formula yields the following expression for the Schur…
LinAlg
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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 inner product of $r$ and the first column of…
Hui Zhang
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Is Schur complement better conditioned than the original matrix?

Consider the following linear system (in block form) with s.p.d. matrix: $$ \begin{pmatrix} A & B\\ B^\top & C \end{pmatrix} \begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} f\\g \end{pmatrix} $$ I'm wondering if elimination of some variables…
uranix
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Does the Schur complement preserve the partial order?

Let $$\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$$ be symmetric positive definite and conformably partitioned matrices. If $$\begin{bmatrix} A_{1} &B_1 \\ B_1'…
Sunni
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Recurrence for characteristic polynomial coefficients

I am trying to find the characteristic polynomial of a $6\times 6$ block matrix $A$ , where each block is a $\ell\times\ell$ matrix, resulting in a $6\ell\times 6\ell$ matrix. The matrix $A$ is defined as: $$A= \begin{bmatrix} U & L & L & L & U &…
Dang Dang
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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{matrix} \right)\cdot \left( \begin{matrix} 2 (a c - b^2) & a d - b…
orangeskid
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What is the linear algebra of the Schur complement?

I am interested in the combinatorics of electrical networks, and this leads naturally into the notion of the Schur complement: Suppose we have a graph (possibly edge-weighted with 'resistances') with "boundary" and "interior" vertices— the idea is…
Eric Stucky
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Minimization of Frobenius Norm and Schur Complement

There is a famous problem Optimization of Frobenius Norm and Nuclear Norm ; however, this is not I want to ask (about proximal operator). Suppose I have an easy optimization problem: $$\min_Q \|Q-Q_N\|_F$$ where $\|\cdot\|_F$ is the Frobenius norm:…
sleeve chen
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Schur complement condition for positive definiteness of operators

To verify if a symmetric block matrix is positive definite, one can check the definiteness of its diagonal blocks and the Schur complement of the respective blocks. Is this also true in the infinite dimensional setting? Precisely, being $A$, $B$ and…
shamisen
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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 $M/A$ and $M/D$ , and the off-diagonal blocks $B$ and $C$ , can…
Benjamin Kay
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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
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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 LMI? Explicitly, can we show something like:…
Morad
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