Tagged Questions
0
votes
0answers
21 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
1answer
34 views
Singular Value Decomposition of a block diagonal matrix
For a block diagonal matrix, we have an identity for its cholesky decomposition i.e.
$chol(Z) = chol(blockdiag(A,B,...)) = blockdiag(chol(A),chol(B),...)$
(Here, $Z = blockdiag(A,B,...)$)
I want to ...
7
votes
1answer
60 views
*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$
I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form
$A\in ...
0
votes
1answer
26 views
proof about deteminant of a complex linear transformation
say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$
such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$
I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
1
vote
2answers
58 views
deteminant of a block skew-symmetric matrix
If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix}
how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix}
so the determinant is ...
5
votes
5answers
136 views
How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$
Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix
$$\mathbf{M} = \begin{pmatrix}0 & \mathbf{B} \\ \mathbf{A} & 0\end{pmatrix}.$$
How to prove that ...
1
vote
1answer
28 views
Proving properties of triangular matrices
This is the question that I'm having trouble with:
I understand what the alternating and multilinear properties are, and I know that the determinant of the matrix is an alternating, multilinear ...
3
votes
1answer
60 views
Fast way to calculate determinant for a block matrix
I have a block matrix
$$Q_{(n+m-1)\times(n+m-1)} = \begin{pmatrix} A & -J\\-J^t & B \end{pmatrix}$$
where
$$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
0
votes
2answers
69 views
Linear algebra: need help with proof
Can someone please help me with this proof.
For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
12
votes
2answers
137 views
Block Diagonal Matrix Diagonalizable
I am trying to prove that:
The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable.
If $A\in\mathbb{C}^n$ ...
2
votes
1answer
50 views
Jordan basis of $A$ when $A$ is the companion matrix?
The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is:
$$
...
0
votes
1answer
47 views
Rewriting a quadratic Matrix equation as a quadratic vector equation
Consider the set of $N \times N$ matrices $\{W_i\}_{i=1}^{i=L}$, set of $N \times 1$vectors $\{g_i\}_{i=1}^{i=L}$ and $\{h_i\}_{i=1}^{i=L}$. Now consider the following sum
\begin{align}
...
2
votes
1answer
79 views
Block diagonalizing a real matrix
I need to prove that for every real linear operator $T:\mathbb{R}^n\longrightarrow \mathbb{R}^n$, there exists an orthonormal basis of $\mathbb{R}^n$ such that the corresponding matrix is block ...
0
votes
0answers
20 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
2answers
92 views
How do I write this matrix in Jordan-Normal Form
I have the matrix $A=\begin{pmatrix}2&2&1\\-1&0&1\\4&1&-1\end{pmatrix}$, I want to write it in Jordan-Normal Form. I have $x_1=3,x_2=x_3=-1$ and calculated eigenvectors ...
2
votes
2answers
39 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 $M=AB^T$, ...
3
votes
1answer
63 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 ...
2
votes
3answers
183 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 ...
2
votes
1answer
106 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 ...
1
vote
2answers
91 views
Eigenvalues and Jordan form
I have a $5\times 5$ matrix and I need to find the Jordan form and its inverse. I know how to find the inverse. But for the Jordan form I am screwed.
The matrix is $$\begin{bmatrix}3 & 0 & 0 ...
4
votes
1answer
66 views
Sum of elements of inverse matrix
Assume NxN matrix A of complex values. I want to calculated the sum of all elements of its inverse. Does anybody have any good idea how to do this? The problem is that calculating the inverse is ...
0
votes
1answer
20 views
How to transform this matrix & swap its columns?
I'm looking for a transformation matrix (or set of transformation matrices) that transforms matrix $\mathbf A = \begin{pmatrix} a&b&i&j\\ c&d&k&l \\ e&f&m&n \\ ...
3
votes
1answer
59 views
How to “flip” and change the sign of one particular row of this matrix?
I would like to transform the following matrix :
$\mathbf A$ =$\ \begin{bmatrix}
a&b\\
c&d\\
e&f\\
g&h
\end{bmatrix}\ $ into this one : $\mathbf B$ = $\ \begin{bmatrix}
g&-h\\
...
-1
votes
2answers
90 views
2
votes
1answer
113 views
The rank of a block matrix as a function of the rank of its submatrices.
I would like to post this problem here in this forum.
Having the following block matrix:
\begin{equation}
M=\begin{bmatrix}
S_1 &C\\
C^T &S_2\\
\end{bmatrix}
\end{equation}
I would like to ...
1
vote
0answers
75 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 ...
4
votes
1answer
98 views
Eigenvector of a sparse structured matrix corresponding to the eigenvalue 1
I have a matrix with the following sparsity pattern:
$M =
\begin{bmatrix}
\ast &\ast &0 &0 &0 &0 &0 &0\\
0 & 0 &\ast &\ast &0 &0 &0 &0 \\
0 ...
0
votes
1answer
73 views
show blockmatrix is invertible
Let $B \in \mathbb{R}^{n \times n}, C \in \mathbb{R}^{m \times n}, m \leq n$ and $\operatorname{rank} C = m$
Suppose for every $v \neq 0$ with $Cv=0$ it is $v^TBv > 0$.
Show: then $A = ...
0
votes
4answers
68 views
Determinant of a $4\times4$ matrice with one unknown?
I have to calculate the determinant of this matrice. I want to use the rule of sarrus, but this does only work with a $3\times3$ matrice:
$$
A=
\begin{bmatrix}
1 & -2 & -6 & u \\
...
2
votes
3answers
109 views
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 ...
1
vote
0answers
55 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 ...
3
votes
0answers
96 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 ...
1
vote
1answer
48 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 ...
2
votes
0answers
92 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 ...
1
vote
3answers
181 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 ...
2
votes
1answer
283 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
72 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:
$${\mathbf M} = \begin{bmatrix}\mathbf A_{11} & \dots & \mathbf A_{1n} \\ \vdots & \ddots & ...
4
votes
0answers
59 views
Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?
I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants):
$$
\log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
1
vote
3answers
165 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 & ...
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 $e^{X'}$ ...
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 ...
