All Questions
Tagged with matrices block-matrices
61 questions
15
votes
3
answers
3k
views
Determinant of a $k \times k$ block matrix
Consider the $k \times k$ block matrix:
$$C = \left(\begin{array}{ccccc} A & B & B & \cdots & B \\ B & A & B &\cdots & B \\ \vdots & \vdots & \vdots & \...
- 301
4
votes
1
answer
594
views
Smith Normal Form for block matrices over the integers
Are there any known results on the Smith Normal Form for block matrices over the integers?
In particular, I am interested in matrices of size $kr \times ks$ made of square blocks of size $k$ such that ...
- 41
0
votes
1
answer
1k
views
How do eigenvalues change if we duplicate a row and column of a symmetric matrix?
Let $\bf A$ be a $n \times n$ symmetric positive semidefinite matrix whose first column is denoted by ${\bf a}_1$. We define a new matrix,
$$ {\bf B} = \begin{bmatrix} a_{11} & {\bf a}_1^T \\ {\bf ...
17
votes
1
answer
735
views
A matrix completion problem
In their paper, Corners of normal matrices, Rajendra Bhatia and Man-Duen Choi asked the following question:
Given a matrix pair $(B,C)$ where $B,C∈M_n$, does there exist matrices $A,D ∈ M_n$ such ...
- 3,699
2
votes
1
answer
1k
views
Form of a block upper triangular matrix of finite order
If I take a diagonalizable block upper triangular matrix whose diagonal blocks are of finite order, is it true that away from the leading block diagonal, the matrix is zero?
I think the statement is ...
- 21
3
votes
0
answers
549
views
Canonical forms for block-positive-definite matrices
Suppose we are given a block $2\times 2$ matrix that is positive-definite, and let's suppose for simplicity that the blocks along the main diagonal are the identity. So
$$
\begin{bmatrix} I & X \\\...
- 427
5
votes
1
answer
2k
views
Condition for block symmetric real matrix eigenvalues to be real
I have a $2n \times 2n$ block symmetric matrix that in the simplest case ($n=2$) looks like:
$$
M_2 = \begin{bmatrix}
a_1 & 0 & b_{1,2} & -b_{1,2}\\\
0 & -a_1 & b_{1,2} & -b_{...
- 151
2
votes
2
answers
242
views
Integer square $2 \times 2$ block matrix inverse
Let $\mathbf{M}$ be an integer square $2 \times 2$ block matrix
$$
\mathbf{M} =
\left(
\begin{array}{cc}
\mathbf{A} & \mathbf{B} \\
\mathbf{C} & \mathbf{D}
\end{array}
\right) ,
$$
where $\...
- 61
7
votes
2
answers
3k
views
Factorizing a block symmetric matrix
Let $X,Y\in\mathbb{R}^{n\times n}$ be symmetric matrices. You may assume that $X$ is positive semidefinite and $Y$ negative semidefinite, if needed, but not that they are invertible.
I would like to ...
- 20.2k
16
votes
6
answers
13k
views
Showing block diagonal structure of matrix by reordering
Suppose we have a block-diagonal matrix $M$, but the block diagonal structure is not immediately apparent from looking at the matrix because the rows/columns are shuffled.
I wish to find a reordering ...
- 911
5
votes
2
answers
2k
views
Iterated calculation of determinants
Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...
- 11.1k