For questions about matrices which are defined block wise, like $\pmatrix{A&B\ C&D}$ where $A,B,C$ and $D$ are themselves matrices. Use this tag with (matrices), and often with (linear-algebra).
Questions tagged [block-matrices]
969 questions
44
votes
3 answers
Prove that the eigenvalues of a block matrix are the combined eigenvalues of its blocks
Let $A$ be a block upper triangular matrix:
$$A = \begin{pmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{pmatrix}$$
where $A_{1,1} ∈ C^{p \times p}$, $A_{2,2} ∈ C^{(n-p) \times (n-p)}$. Show that the eigenvalues of $A$ are the combined eigenvalues of…
tsiki
- 749
40
votes
7 answers
Determinant of a block lower triangular matrix
I'm trying to prove the following: Let $A$ be a $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then,
$$\det\left(\begin{array}{cc}
A&0\\
C&D
\end{array}\right) = \det(A)\det(D).$$
Can I just say that $AD - 0C =…
Buddy Holly
- 1,209
27
votes
3 answers
Determinant of a block upper triangular matrix
How prove the following equality for a block matrix?
$$\det\left[\begin{array}[cc]\\A&C\\
0&B\end{array}\right]=\det(A)\det(B)$$
I tried to use a proof by induction but I'm stuck. Is there a simpler method? Thanks for help.
user66407
26
votes
2 answers
How to denote matrix concatenation?
Trivial question: Is there any standard notation for the concatenation of two or more matrices?
Example:
$$A = \left(\begin{array}[c c]
- a_1 & a_2\\
a_3 & a_4
\end{array}\right),$$
$$B = \left(\begin{array}[c c]
- b_1 & b_2\\
b_3 & b_4
…
Anibal Troilo
- 263
26
votes
1 answer
Principal submatrices of a positive definite matrix
Let $\mathbf{A}\in\mathbb{R}^{N \times N}$ be symmetric positive definite. For some $1\leq k
Drew
- 451
25
votes
4 answers
Block diagonal matrix diagonalizable
I am trying to prove that:
The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diagonalizable, if only if $A$ and $B$ are diagonalizable.
If $A\in GL(\mathbb{C}^n)$ and $B\in GL(\mathbb{C}^m)$ are diagonalizable, then…
FASCH
- 1,762
- 1
- 21
- 31
25
votes
4 answers
Inverse of a $2 \times 2$ block matrix
Let
$$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}}$$
But, what if…
Claire
- 421
24
votes
7 answers
What is the codimension of matrices of rank $r$ as a manifold?
I'm reading through G&P's Differential Topology book, but I hit a wall at the end of section 4. There is a result stating
The set $X=\{A\in M_{m\times n}(\mathbb{R}):\mathrm{rk}(A)=r\}$ is a submanifold of $\mathbb{R}^{m\times n}$ with codimension…
Geovanna Anthony
- 669
21
votes
1 answer
Inverse of a block matrix with singular diagonal blocks
I have a special case where $$X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$$ and:
$X$ is non-singular
$A \in \Bbb R^{n \times n}$ is singular
$B \in \Bbb R^{n \times m}$ is full column rank
$C\in \Bbb R^{m \times n}$ is full row…
Shyam
- 353
- 1
- 2
- 12
21
votes
2 answers
How to find the eigenvalues of a block-diagonal matrix?
The matrix $A$ below is a block diagonal matrix where each block $A_i$ is a $4 \times 4$ matrix with known eigenvalues.
$$A= \begin{pmatrix}A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_n …
cgo
- 1,970
20
votes
3 answers
Proofs of determinants of block matrices
I know that there are three important results when taking the Determinants of Block matrices
$$\begin{align}\det \begin{bmatrix}
A & B \\
0 & D
\end{bmatrix} &= \det(A) \cdot \det(D) \ \ \ \ & (1) \\ \\
\det \begin{bmatrix}
A & B \\
C & D…
Perturbative
- 13,656
19
votes
1 answer
The inverse of a block-upper triangular matrix
Is it true that $$\begin{pmatrix} A & * \\ 0 & B \\ \end{pmatrix}^{-1} = \begin{pmatrix} A^{-1} & * \\ 0 & B^{-1} \\ \end{pmatrix}$$
where $A$ and $B$ are $m \times m$ and $n \times n$ invertible, and * is for unspecified blocks ?
emelie
- 449
18
votes
0 answers
Matrix diagonalization theorems and counterexamples: reference-request.
I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra.
In this question I understand the question of matrix diagonalization very broadly:
suppose we have some group $G$ of matrices, where $G$ is one…
Fiktor
- 3,192
18
votes
0 answers
Bounding the minimum singular value of a block triangular matrix
Question:
What is the sharpest known lower bound for the minimum singular value of the block triangular matrix
$$M:=\begin{bmatrix}
A & B \\ 0 & D
\end{bmatrix}$$
in terms of the properties of its constituent matrices?
Motivation:
Block triangular…
Nick Alger
- 19,977
18
votes
4 answers
General expression for determinant of a block-diagonal matrix
Consider having a matrix whose structure is the following:
$$
A =
\begin{pmatrix}
a_{1,1} & a_{1,2} & a_{1,3} & 0 & 0 & 0 & 0 & 0 & 0\\
a_{2,1} & a_{2,2} & a_{2,3} & 0 & 0 & 0 & 0 & 0 & 0\\
a_{3,1} & a_{3,2} & a_{3,3} & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 0…
Andry
- 1,053