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.
Questions tagged [schur-complement]
134 questions
25
votes
4 answers
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
- 421
12
votes
1 answer
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
- 1,928
11
votes
3 answers
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…
Mårten W
- 3,550
- 4
- 27
- 42
10
votes
1 answer
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 the…
Hui Zhang
- 606
10
votes
1 answer
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
- 20,093
9
votes
1 answer
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
- 7,773
- 1
- 22
- 57
7
votes
3 answers
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
$$\begin{bmatrix}
A_{1} &B_1 \\…
Sunni
- 4,586
5
votes
1 answer
Recurrence for characteristic polynomial coefficients
I am trying to find the characteristic polynomial of a $6\times 6$ block matrix $A$, where each block is a $\ell\times\ell$ matrix, resulting in a $6\ell\times 6\ell$ matrix. The matrix $A$ is defined as:
$$A= \begin{bmatrix} U & L & L & L & U & U\\…
Dang Dang
- 320
5
votes
0 answers
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
- 56,630
5
votes
1 answer
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…
sleeve chen
- 8,576
5
votes
1 answer
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 $A$, $B$…
shamisen
- 870
4
votes
1 answer
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 I…
Benjamin Kay
- 151
4
votes
1 answer
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:
$$\lambda_{\min}(M) \ge…
Probabilist
- 418
4
votes
1 answer
Can $X - Y A^\dagger Y^T\succ0$ be written as an LMI where $A^\dagger$ is a pseudoinverse?
I have the constraint
\begin{align}
X - Y A^\dagger Y^T\succ0,
\end{align}
where $A^\dagger$ is the pseudoinverse of $A\succeq0$. Can we still use the Schur complement to write the constraint as an LMI?
Explicitly, can we show something…
Morad
- 690
4
votes
1 answer
Positive semidefiniteness of block matrix when diagonal blocks are not invertible
Let
$$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 positive semi-definite?
For example, consider the case…
zola
- 49
- 4