# Tagged Questions

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).

2answers
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### Eigenvalue of anti triangular block matrix (skew matrix?)

I have an real anti-triangular matrix $M=\left[ \begin{array}{cc} A & B \\ I & 0 \\ \end{array} \right]$ where I is an identity matrix. $A$, $B$, $I$, $0$ are all square real ...
0answers
150 views

### when block matrix is positive definite ,

can one tell me when the block matrix M=[A B,C D] is positive definite such as: 1-the four block A,B,C and D are symmetric diagonal positive definite matrices 2-M is asymmetric, so C is not the ...
2answers
64 views

### represent the matrix into rank 2

Given an $n\times 1$ vector $x$ and an $n\times 1$ vector $y$. The $n\times n$ matrix $xy^T$ is a rank one matrix. Now let $M=xy^T+yx^T$, how do we represent the matrix $M$ as a rank 2 form $M=AB^T$, ...
1answer
91 views

### decomposing PSD block matrix into two PSD block matrices

Given $Q = \left( \begin{array}{ccc} A + B & C \\ C^T & D\end{array} \right)$, where we know that $Q, A, B, D$ are all positive semi-definite, square, but not necessarily equally sized ...
1answer
52 views

### Block diagonalization of a symmetric square boolean matrix

I have a symmetric square matrix with elements from $\{0,1\}$. How can I block diagonalize it only swapping lines and columns or detect it's not possible?
3answers
961 views

### Prove that the rank of a block diagonal matrix equals the sum of the ranks of the matrices that are the main diagonal blocks.

\begin{equation*} X= \begin{pmatrix} A& 0 \newline 0& B \end{pmatrix} \end{equation*} If A and B are some matrices and 0 is a zero matrix, prove that $\ rank(X)=\ rank(A)+\ rank(B)$. Also, if ...
1answer
96 views

### Index of nilpotency jordan block

If $T$ is an endomorphism, there exists a basis, according to which $T$ will be a block-diagonal matrix. Because if $V$ is the domain of $T$, $V$ will be the direct sum of the generalized eigenspaces, ...
1answer
794 views

### Is always possible to find a generalized eigenvector for the Jordan basis M?

$A$ is a defective matrix, meaning that there are fewer linearly independent eigenvectors than eigenvalues; the algebraic multiplicity of $\lambda_1$ is $v_i = 2$ while the geometric multiplicity is ...
2answers
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3answers
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### Quadratic equation with matricial coefficients

If I have a equation in the form $${\lambda ^2}{I_N} + \lambda {M_1} + {M_2} = {0_N}$$ where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
2answers
145 views

### Eigenvalues of Block matrices with known eigenvalues

Let's have the following $(n+1) \times (n+1)$ matrix with block elements $\mathbf{Y} = \begin{bmatrix} \mathbf{A} & - \mathbf{w} \\ - \mathbf{w}^{T} & b \end{bmatrix}$ where $\mathbf{A}$ is ...
1answer
175 views

### Sorting Matrix to Block structure

I have a symmetric matrix and I want it to be as block-like as possible. I don't have a clear definition. I want the smallest number of groups of non zero elements or maybe the most non-zero elements ...
0answers
160 views

### What kind of matrix/tensor notation is this?

I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues. About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
3answers
530 views

### Determinants of block matrices

Let $A,B \in \mathbb{R}^{n,n}$. Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$. Show that $\det(C) \in \mathbb{R}$ ...
1answer
56 views

### Help regarding a weird Matrix

Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
0answers
129 views

### Properties about Matrices that can be proved by only using Block Multiplication of Matrices

I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
1answer
71 views

1answer
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### Is eigenvectors/eigenvalues of symetric matrix related with eigenvectors/eigenvalues of its block?

$A$ is square symmetric matrix as: $$A=\begin{pmatrix} A_1&A_2 \\ A_3 & A_4 \end{pmatrix}$$ I have two points which need help to understand clearly: All blocks $A_1$, $A_2$, $A_3$, $A_4$ ...
0answers
70 views

2answers
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2answers
264 views

### rank one update

Given a matrix $X$, we can compute its matrix exponential $e^X$. Now one entry of $X$ (say $x_{i,j}$) is changed to $b$, the updated matrix is denoted by $X'$. My problem is how to compute $e^{X'}$ ...
1answer
70 views

### BIBD problem, combinatorics

How many different (7,7,3,3,1) BIBD are there on vertex set[7]? Note that two such designs are different if their sets of blocks are different. That is, we do not require that the BIBDs be ...
1answer
134 views

### block design question combinatorics

Let D be a BIBD with parameters (b,v,r,3,1). Prove that either v=6i+1 or v=6i+3 for some non-negative integer i. BIBD stands for a Balanced Incomplete Block Design My work, theorem states that bk=vr ...
1answer
76 views

### bilinear equation OR diagonal matrix search

Having a real square $n \times n$ symmetric matrix $B$ (which consists of 2 square blocks of positive numbers and 2 rectangular blocks of negative numbers) I want a real diagonal matrix $D$ such that ...