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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vote
1answer
41 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
49 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 ...
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votes
1answer
76 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:
$${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & ...
4
votes
0answers
60 views
Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?
I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants):
$$
\log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
3
votes
0answers
69 views
+50
special case of the inverse eigenvalue problem
I need to construct a non-negative matrix with desired eigenvalues. To that end, I came up with a block matrix of the following form:
$$
\mathbf{M} = \begin{vmatrix}
\mathbf{A} & \mathbf{b} \\
...
3
votes
0answers
99 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
97 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
78 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
53 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
76 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
61 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
0answers
83 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
62 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
15 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 ...
0
votes
0answers
28 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
23 views
Linear least squares decomposition of a submatrix of a gaussian random matrix
Suppose $X$ is an $n\times p$ random matrix whose rows are picked iid according to $N(0,\Sigma)$, $\Sigma$ being $p\times p$ symmetric, positive definite. Let $S,T\subset \{1,2,\ldots,p\}$ be such ...
0
votes
0answers
42 views
Perturbing/sensitivity of the eigenvalues of a block tridiagonal matrix
This question is related to my former question:
Checking if one "special" kind of block matrix is Hurwitz
Given the next matrix
$$
J = \begin{bmatrix}-(B+B^T) & B \\ 0 ...
0
votes
0answers
59 views
on norm of submatrix of the inverse and inverse of a submatrix
Given an M-matrix, say $M\in\mathbb{R}^{n\times n}$, which in block form is
$M$ = \begin{pmatrix}
A & B\\
C & D
\end{pmatrix}
where $A\in\mathbb{R}^{k\times k}$ and ...
0
votes
0answers
174 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} ...
