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]

735 questions

**38**

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

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**35**

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

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**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

**23**

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

- 1,632
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**20**

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

- 203
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**19**

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

- 333
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- 2
- 11

**17**

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

- 331
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- 9

**16**

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

- 2,254
- 17
- 19

**16**

votes

**6**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

- 569
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- 10

**16**

votes

**2**answers

### The determinant of block triangular matrix as product of determinants of diagonal blocks

I am given the following partitioned, upper-triangular matrix:
$$
\begin{bmatrix}
A_1 &* &* &* &* &* \\
0& A_2 &* &* &* &* \\
.& 0& A_3 &* &* &* \\
.& 0& 0 &... &* &. \\
.& 0& 0& 0& ... &. \\
0& .& ...& 0&0 &…

Dor Shalom

- 633
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- 14

**16**

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

- 16,798
- 11
- 59
- 85

**14**

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

- 1,618
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- 14
- 28

**13**

votes

**4**answers

### eigenvalues of certain block matrices

This question inquired about the determinant of this matrix:
$$
\begin{bmatrix}
-\lambda &1 &0 &1 &0 &1 \\
1& -\lambda &1 &0 &1 &0 \\
0& 1& -\lambda &1 &0 &1 \\
1& 0& 1& -\lambda &1 &0 \\
0& 1& 0& 1& -\lambda &1 …

Michael Hardy

- 1
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**13**

votes

**3**answers

### Why does a diagonalization of a matrix B with the basis of a commuting matrix A give a block diagonal matrix?

I am trying to understand a proof concerning commuting matrices and simultaneous diagonalization of these.
It seems to be a well known result that when you take the eigenvectors of $A$ as a basis and diagonalize $B$ with it then you get a block…

vonjd

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**13**

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

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