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).

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3
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2answers
56 views

Basis of Kernel of a matrix

Given $\theta>0$. Let $H$ be $5 \times 6$ matrix $$\left[\begin{matrix} 1 & -1 & 0 & 0 & 0 & 0 \\ 1 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & -1 ...
2
votes
1answer
83 views

Congruence of a matrix

Let $X=\begin{bmatrix}a&b&0&0\\ c&d&0&0\\0&0&-a&-b\\0&0&-c&-d\end{bmatrix}$ where $a,b,c,d\in \mathbb{Z}$. For a such given $X$, is there a $4\times ...
1
vote
1answer
44 views

Proving that $M_p(M_q (K)) \cong M_{pq} (K)$.

My textbook finishes the proof of one of the theorems with the following fact: $$ M_p(M_q (K)) \cong M_{pq} (K), $$ where $K$ is a field, and it says that it is true by "block multiplication ...
1
vote
1answer
80 views

Linear Algebra — Block Matrix Inversion

Please excuse my formatting... $X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that ...
1
vote
1answer
201 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 ...
1
vote
1answer
58 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 ...
0
votes
1answer
24 views

Calculating Partitioned Matrices from subs

Say you have a matrix $A$ which is of size $P\times P$ and a number $Q < P$ can be used to take a partition of said matrix, where: $A_1$ is the upper-left sub matrix, with dimension $Q\times Q$, ...
0
votes
1answer
40 views

2nd order Matrix differential equation

$\ddot{X}+W\dot{X}=X$, $W$ is n-dimensional skew symmetric matrix. $X$ is a column vector and $I$ is identity matrix of appropriate dimension. \begin{equation} \left(\begin{array} XX \\ \dot{X} ...
0
votes
1answer
87 views

Iterative Method for a special kind of Sparse Matrix

I've the following problem.I've a sparse square Matrix $\bf M$. I can write $\bf M$ as: $${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & ...
6
votes
0answers
124 views

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 ...
5
votes
0answers
71 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
3
votes
0answers
46 views

Find signature of symmetric block matrix, given the diagonal blocks are positive / negative definite - Check my proof

This may be a basic question, but I'd like someone to double check it. We are given the matrix $A=\begin{pmatrix} A_1 & C \\ C^T & A_2\end{pmatrix}$ where $A_1$ is a $k$ by $k$ positive ...
3
votes
0answers
179 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 ...
2
votes
0answers
41 views

linear algebra formulation help

I asked this on mathoverflow as well and apologies for cross-posting. I am trying to compute this so-called bending energy matrix. The bending energy of a thin plate in 3D is given by: $$ BE = ...
2
votes
0answers
133 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 ...
2
votes
0answers
90 views

Determinant, number of non zero columns

Trying to build a reduction from the maximum coverage problem to my research problem, I'm facing this difficulty : Let $X$ be a $n \times m$ binary matrix (with $m > n$), can we define a square ...
1
vote
0answers
19 views

A question about eigenvalues of a special block matrix

Thanks for anyone who views or answers this question! $N\times N$ matrix $L$ can be partitioned into $p\times p$ blocks, ...
1
vote
0answers
12 views

generalisation of Knonecker matrix product

In the Kronecker matrix product $C = A\otimes B$ we have that $C(i,j)=A(i,j)*B$ where the elements $A(i,j)$ are just numeric scalar values. What if the $A(i,j)$ are matrix operators which act on ...
1
vote
0answers
53 views

Use Frobenius inequality to prove $\mathrm{rank}(A)+\mathrm{rank}(B)\geq\mathrm{rank}(AB)+ \mathrm{rank}(A+B)$

If $A, B$ are $n\times n$ matrices, $AB=BA$, then $$\mathrm{rank}(A)+\mathrm{rank}(B)\geq\mathrm{rank}(AB)+ \mathrm{rank}(A+B)$$ Can one try Frobenius inequality to prove this? we must find three ...
1
vote
0answers
34 views

Separating the eigenvalues of a Hermitian matrix with a special block structure

I have a square matrix $J \in \mathbb{C}^{2n \times 2n}$ where, $J=\begin{pmatrix} A&B \\\bar{B} & A \end{pmatrix}$ $A \in \mathbb{R}^{n \times n}$ and is ${\bf diagonal}$ $B \in ...
1
vote
0answers
23 views

Condition for a block matrix to be positive semi-definite

Let's say we have two positive semi-definite matrices $A$ and $B$ and a negative real $\lambda$. What would be the conditions for the matrix $M$, defined as follows, to be positive semi-definite too ? ...
1
vote
0answers
25 views

Can we find a reduced row-echelon form for these matrices?

Starting with a Vandermonde matrix: $$V = \begin{bmatrix} 1^1 & 1^2 & 1^3 & \dots & 1^n \\ 2^1 & 2^2 & 2^3 & \dots & 2^n \\ 3^1 & ...
1
vote
0answers
32 views

Understanding of a formula with matrix summation

We are a quite a few students at in the class struggling to compute this would anyone be able to help? Also note --> ' means transpose. Sorry for the misunderstanding, when I said that it is not ...
1
vote
0answers
176 views

Proof that if a matrix is invertible, its rank is maximum

I have to prove that if a square matrix $A \in \mathfrak{M}_n (\mathbb{K})$ is invertible, then $rg(A) = n$. The thing is I cannot use vector spaces, subspaces, etc... to prove this, only matrix ...
1
vote
0answers
46 views

Are all columns or rows in a block toeplitz convolution matrix linearly independent?

My question specifically relates to the case where the vector that the matrix blocks were formed from have lower orders than the dimensions of the sub matrices. Consider a vector (filter) $a[k]$ of ...
1
vote
0answers
20 views

DCT and split-radix algorithms

I am studying a paper which describes split-radix algorithms by making matrix factorizations, so that e.g. DCT 8x8 can be computed via 4 DCT4x4. I must apologize, but the question is related to an ...
1
vote
0answers
76 views

Finding correct equivalence matrix for all group representations

This question is linked to a previous one that I asked: Attempting to find a specific similarity (equivalence) matrix I have a group of 24 elements, with two generators. I need to find an ...
1
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0answers
142 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) ...
1
vote
0answers
45 views

Is there any simple way to write down permutations set-theoretically?

Let $\{J_{n_1}(\lambda_1),...,J_{n_m}(\lambda_m)\}$ and $\{J_{l_1}(\mu_1),...,J_{l_k}(\mu_k)\}$ be finite sequences of Jordan blocks with entries in a field $F$. Let $A\triangleq ...
1
vote
0answers
159 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 ...
1
vote
0answers
147 views

Square root of a squared block matrix

I’m trying to compute the square root of the following squared block matrix: \begin{equation} M=\begin{bmatrix} A &B\\ C &D\\ \end{bmatrix} \end{equation} (that is $M^{1/2}$) as function of ...
1
vote
0answers
109 views

How does adding extra row and column of ones affect a matrix's inverse?

I'm working on a homework problem, and I'm stuck. I guess my linear algebra still needs some work... I've arrived at $\mathbf{D}= \left[ \begin{matrix} \mathbf{C} & \mathbf{1}^T \\ \mathbf{1} ...
1
vote
0answers
78 views

Eigen value of a complex block matrix

I have a following block matrix M1 =([a11 a12;a21 a22] [b13;b23];[b13;b23]' d33) + ([c -cr]'*[c -cr] 0;0 I) now what i observe is that whether i use ...
0
votes
0answers
9 views

How to find the rules by which eig(A) = diagonal entries

Suppose A = [a b c;d e f;g h i] how to find a series of rules such that eig(A) = a, e, i such as h = 0 or f = 0? Is there any rule for this? (I know we can find this by principal submatrix, but I ...
0
votes
0answers
65 views

Block Diagonalisation of 4x4 Matrix

I'm attempting to find a 4x4 matrix, P, that will convert my matrices, $A = \begin{bmatrix}1&1&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{bmatrix}$ and, ...
0
votes
0answers
26 views

Block diagram structure function marks and minimal cut sets

I have following block diagram, How can I structure function marks and minimal cut sets? I would like it to be explained if possible so that I can understand it. Thanks.
0
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0answers
28 views

Block Circulant Matrix Inverse

I have a nearly block-tridiagonal "block circulant" matrix of the form $$G=\left(\begin{array}{ccccccc} B & C & & & & & A\\ A & B & C\\ & A & B & C\\ ...
0
votes
0answers
33 views

Can we reduce this matrix to the identity, which contains binomial elements?

We are given a function: $$f(a,b,m) = \binom{n}{b}\binom{n-b}{a}\binom{n-a-b}{m-a}$$ We can suppose we have the following $(n/2)^2 \times (n+1)$ matrix (form), that we wish to find the value for the ...
0
votes
0answers
12 views

Can we prove that this tabular algorithm works correctly?

Finding an answer to the following question is very important, because it will help prove an algorithm works correctly. It is also extremely hard to explain, so I'm hoping that someone will help me ...
0
votes
0answers
29 views

Find out the smallest disk like ($|z-1| < r$ ) in the complex plane containing the eigenvalues of the given matrix

Consider the given matrix $$\left[\begin{matrix}1& -2& 3& -2 \\1& 1& 0& 3\\-1& 1& 1& -1\\0& -3& 1& 1&\end{matrix}\right]$$ Find out the smallest ...
0
votes
0answers
48 views

Finding basis of null space for block matrix

I am trying to determine the full-rank basis $Z \in \mathbb R^{(n+m) \times }?$ for the null space of the matrix $$M= \left( \begin{array}{yy} -A & B \\ x & y \end{array} \right)$$ with ...
0
votes
0answers
106 views

Row & Column Removal and Rank Reduction

I have a problem involving a n x n square, real matrix $K$ which is initially full rank and is not positive definite. In each iteration of my program, I have to remove a row and the corresponding ...
0
votes
0answers
50 views

Can we perform this operation on block matrices?

We have a block matrix: $$ \left[\begin{array}{c|c|c} A & 0 & 0 \\ \hline 0 & B & 0 \\ \hline 0 & 0 & C \end{array}\right] $$ Here $A$, $B$ and $C$ are all permutation ...
0
votes
0answers
75 views

Pseudo-inverse of something similar to Kronecker product

I have a matrix $C$ that looks like a Kronecker product of two matrices $A$ and $B$, but not quite. $A$ is a $p\times q$-matrix and $B$ is a $r\times q$-matrix, and $C$ looks like this: $$C = ...
0
votes
0answers
51 views

Equivalence theorem between JNF & Companion matrices

We treated both concepts in class and now I was wondering whether the number of companion matrices in the Frobenius Normalform is equal to the number of Jordan blocks? It is just an assumption and I ...
0
votes
0answers
81 views

Need to show summation is positive definite

I want the following square matrix to be positive definite (or have no eigenvalues equal to zero, or have a nonzero determinant) for all $N$ \begin{equation*} X = \sum\limits_{k=0}^N ...
0
votes
0answers
17 views

Anti-Blockdiagonalizing Bipartite Adjacence Matrices

I'm currently facing the following problem: Given an adjacence matrix $A$ of a bipartite graph on $2n$ (black and white) vertices. How can I get an anti-block diagonal version of the adjacence ...
0
votes
0answers
28 views

Basis transformation of frobenius normalform

If I have an endomorphism $A$ with characteristic polynomial $p=(\lambda-a_1)^k\cdot...\cdot(\lambda-a_i)^j$. This determines our blocks in the frobenius normal form(but not the order of the companion ...
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0answers
87 views

Quadratic form of block matrix

If one has a block matrix $\tilde A = \left[ {\begin{array}{*{20}{c}} D&{{0_{n \times n}}}\\ {{0_{n \times n}}}&{{0_{n \times n}}} \end{array}} \right]$ where $D\in {R^{n \times n}}$ is a ...
0
votes
0answers
192 views

Eigen values of sum of two matrices

I have a following block matrix \begin{align*} M_1 &= M_a + M_b\\\ M_a &= \begin{pmatrix} a_{11} & 0 &b_{13}\\\ 0 & a_{22}& b_{23}\\\ b_{13}^T & b_{23}^T & d_{33} ...