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).
1
vote
3answers
182 views
Positive-definiteness of block diagonal matrix
Given a block diagonal matrix $A$ like so:
$$A=
\begin{bmatrix}
B & & \\
& C & \\
& & D \\
\end{bmatrix}
$$
Given $B$ is ...
5
votes
2answers
209 views
Eigenvalue decomposition of block covariance matrix for Canonical Correlation Analysis (CCA)
Edited:
My question is related to a tutorial I was reading.
The covariance matrix is a block matrix where $C_{xx}$ and $C_{yy}$ are within-set covariance matrices and $C_{xy} = C_{yx}^T$ are ...
2
votes
1answer
287 views
Are there any decompositions of a symmetric matrix that allow for the inversion of any submatrix?
I am given a $J \times J$ symmetric matrix $\Sigma$. For a subset of $\{1, ..., J\}$, $v$, let $\Sigma_{v}$ denote the associated square submatrix. I am in need of an efficient scheme for inverting ...
2
votes
0answers
77 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
81 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} ...
0
votes
1answer
74 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:
$${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & ...
1
vote
1answer
69 views
Is eigenvectors/eigenvalues of symetric matrix related with eigenvectors/eigenvalues of its block?
$A$ is square symmetric matrix as:
$$A=\begin{pmatrix} A_1&A_2 \\ A_3 & A_4 \end{pmatrix}$$
I have two points which need help to understand clearly:
All blocks $A_1$, $A_2$, $A_3$, $A_4$ ...
4
votes
0answers
59 views
Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?
I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants):
$$
\log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
2
votes
1answer
119 views
Spectral radius and powers of a $2\times 2$ block matrix
The problem I'm struggling is the following:
Let $n$ be a positive integer and let $A=%
\begin{pmatrix}
B & C\\
C & B
\end{pmatrix}
\in\mathcal{M}_{2n}(\mathbb{R}_{+})$, where ...
0
votes
0answers
164 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} ...
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 ...
1
vote
3answers
166 views
Find Elementary Matrics E1 and E2 such that $E_2E_1$A = I
I am studying Linear Algebra part-time and would like to know if anyone has advice on solving the following type of questions:
Considering the matrix:
$$A = \begin{bmatrix}1 & 0 & \\-5 & ...
0
votes
2answers
93 views
Eigenvalue of 'extended block' matrix
Thanks guys for the previous answer, Now suppose if I have a matrix e.g
$$M_1 = \begin{pmatrix} \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} & B \\ B' & D ...
5
votes
2answers
176 views
Eigen value of a block matrix
I have following block matrices:
$$M_1 = \left(\begin{array}{cc}A & B\\B' & D\end{array}\right)$$
and
$$M_2 = \left(\begin{array}{cc}A & -B\\-B' & D\end{array}\right)$$
I want to ...
5
votes
1answer
429 views
Eigenvalues of certain block hermitian matrix
Suppose I have a special block, Hermitian matrix $H = \begin{bmatrix} A & B \\ B^* & A^* \end{bmatrix}$, where * denotes conjugate transpose. The blocks $A$ and $B$ are themself Hermitian in ...
5
votes
1answer
842 views
Eigenvalues of block matrices
Let $K$ be a field of characteristic 0, and consider the following block matrix
$$M=\left(\begin{array}{cc} A & B\\ -B&D\end{array}\right),$$
where each block is an $n\times n$ matrix with ...
1
vote
2answers
191 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 $e^{X'}$ ...
1
vote
1answer
59 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 ...
1
vote
1answer
107 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 ...
0
votes
1answer
65 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 ...
