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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1answer
59 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 ...
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 ...
2
votes
1answer
260 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
45 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
89 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
59 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
34 views

most general form of $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution $X = f(A,B,C)$?

What is the most general form of the cubic matrix equation $X - A = X^{-1}B (X^{-1}BX^{-1}+ C)^{-1}$ that has a real solution of the form $X = f(A,B,C)$, where $A,B$ and $C$ are positive definite ...
0
votes
1answer
14 views

Characteristic polynomial of triangular blocks matrix

Let A be a triangular blocks matrix (the blocks are: A1,...,Ak). Show that CA(t)=CA1(t)*...*CAk(t). Any help ? thanks ;) (edit: CA and CAj are the characteristic polynomials of the blocks)
6
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0answers
149 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
73 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
207 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
25 views

Canonical Form of Nilpotent Matrices

Given the matrix $$\hat{S}=\begin{bmatrix} S & *& *&* \\ 0& S &* &* \\ 0& 0& S &* \\ 0&0&0&S\\ \end{bmatrix} $$ where $S$ is an $n \times n$ ...
2
votes
0answers
44 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
139 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
91 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
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0answers
23 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
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0answers
13 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
39 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
41 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
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0answers
29 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
232 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
59 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
28 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
78 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
vote
0answers
156 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
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0answers
164 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
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0answers
169 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
112 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
83 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
7 views

Computing a centralizer in an orthogonal group

Consider a skew-symmetric $(4n+2) \times (4n+2)$ block-diagonal real matrix $A$ in normal form: $$A = \begin{bmatrix} \Lambda_1 & 0 & \cdots & 0\\ 0& \Lambda_2 & \cdots & 0\\ ...
0
votes
0answers
7 views

Solving Toeplitz plus Diagonal System, how should I make use of the structure?

I learned that a Toeplitz system, $Ax = B$ where $A$ is Teoplitz, can be solved in $O(n \log n)$ time using Superfast method. or approximate $A$ similar to Approximation method. I am keep ...
0
votes
0answers
28 views

Computing characteristic polynomial of tridiagonal block matrices

I want to compute the characteristic polynomial of symmetric matrices of the form \begin{bmatrix} A & U & & & 0\\ U & B & V & &\\ & V& C& W &\\ ...
0
votes
0answers
16 views

Block Circulant matrix

Let \begin{align} A=\left(\begin{array}{cc} B & C \\ -C & B \end{array}\right), \end{align} where $B=\left(\begin{array}{ccc} \alpha^R & \beta^R & \gamma^R \\ \gamma^R &\alpha^R ...
0
votes
0answers
18 views

Block Diagonalization related to Direct Sum and Single Eigenvalue?

I'm just a beginner in Linear Algebra, and I've proved myself the following: A matrix $A^{n \times n}$ is block diagonalizable if and only if the base field $F^n$ can be divided into at least two ...
0
votes
0answers
21 views

Design feedback control law to make the whole matrix Hurwitz

Suppose $(A_1, B_1)$ and $(A_2, B_2)$ are both stabilizable. Then we know that we can find some $K_1$ and $K_2$ to make $A_1+B_1K_1$ and $A_2+B_2K_2$ Hurwitz, respectively. Now, for non-zero constant ...
0
votes
0answers
10 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
109 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
28 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
69 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
34 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
13 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
33 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
56 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
109 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
51 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
89 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
84 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
198 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} ...