Tagged Questions
1
vote
1answer
26 views
Nonsingularity of a block matrix
Let $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
and:
If $X$ is non-singular, is $A$ non-singular when $B$ is full column rank and $C$ is full row rank?
0
votes
2answers
44 views
Eigenvalues of a block matrix
For $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
, how are eigenvalues of X
related to eigenvalues of A
?
7
votes
1answer
123 views
Inverse of a block matrix
I have a special case where $X=\left(\begin{array}{cc}
A & B\\
C & 0
\end{array}\right)$
and:
$X$ is non-singular
$A$ is singular
$B$ is full column rank
$C$ is full row rank
How do you ...
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 ...
1
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
...
3
votes
1answer
158 views
Rank of a block-triagonal matrix
Given a matrix $C=\left [ \begin{matrix} A & 0 \\ B & A \end{matrix} \right ]$, where rank(A+B)=rank(B), and rank(B)>rank(A), does rank(C)=rank(A)+rank(B) hold?
A,B are Laplacian matrices.
0
votes
1answer
55 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
64 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
28 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
61 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 ...
1
vote
1answer
29 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
62 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
$$A_{(m-1)\times(m-1)} = n*I_{(m-1)\times(m-1)} \text{ and } B_{n\times n} = ...
0
votes
2answers
71 views
Linear algebra: need help with proof
Can someone please help me with this proof.
For $A,B$ ∈ $F^{n×n}$, show that $AB$ and $BA$ have the same characteristic polynomial.
11
votes
2answers
165 views
Block Diagonal Matrix Diagonalizable
I am trying to prove that:
The matrix $C = \left(\begin{smallmatrix}A& 0\\0 & B\end{smallmatrix}\right)$ is diaginalizable, if only if $A$ and $B$ are diagonalizable.
If $A\in\mathbb{C}^n$ ...
2
votes
1answer
58 views
Jordan basis of $A$ when $A$ is the companion matrix?
The actual question: when $A$ is the companion matrix, why the general form of $M_i$ (the group of columns of the Jordan matrix $M$ that belongs to the block associated to $\lambda_i$) is:
$$
...
0
votes
1answer
50 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}
...
1
vote
1answer
36 views
Find a complex matrix that its square is a sum of a scalar matrix and a jordan block
Attempting to solve a question from homework, relating to Jordan normal form:
Let $a \in \mathbb{C}, N \in M_n^\mathbb{C}$. $N$ is nilpotent of index 3.
Prove that the matrix $S \in ...
2
votes
1answer
84 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 ...
2
votes
2answers
43 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
66 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 ...
1
vote
1answer
75 views
Index of nilpotency jordan block
If $T$ is an endomorphism, there exists a basis, according to which $T$ will be a block-diagonal matrix. Because if $V$ is the domain of $T$, $V$ will be the direct sum of the generalized eigenspaces, ...
1
vote
2answers
101 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
75 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 ...
1
vote
1answer
92 views
Matrix identity for symmetric block matrix
I have a problem proving a matrix identity that comes from a data validation and reconciliation problem. So far, we could "see" numerically that the following identity seems to hold, but could not ...
2
votes
1answer
130 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
77 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
103 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 ...
1
vote
2answers
81 views
Finding the trace of a block matrix
Let $A$ be a $5 \times 5$ skew-symmetric matrix with entries in $\mathbb{R}$ and $B$ be the $5 \times 5$ symmetric matrix whose $(i, j)^{th}$ entry is the binomial coefficient $\binom{i}{j}$ for $1 ...
0
votes
1answer
74 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 = ...
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 ...
5
votes
3answers
353 views
Determinants of block matrices
Let $A,B \in \mathbb{R}^{n,n}$.
Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$.
Show that $\det(C) \in \mathbb{R}$ ...
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 ...
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 ...
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 ...
1
vote
1answer
68 views
Why a (general) block matrix does not have the same eigenvalues of the first block?
Based on the next relation:
$$\det\begin{bmatrix}A & B \\ C & D\end{bmatrix} = \det(A)\det(D - CA^{-1}B),$$
I have that for computing the eigenvalues of the block matrix:
...
3
votes
1answer
122 views
Checking if one “special” kind of block matrix is Hurwitz
I have the next block matrix
$$
J = \begin{bmatrix}A & B \\ K &0\end{bmatrix}
$$
all matrices are square, where $A < 0$ (definite negative), $B$ has all its eigenvalues with positive real ...
1
vote
3answers
187 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
238 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 ...
1
vote
0answers
84 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
3answers
177 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:
$$A = \begin{bmatrix}1 & 0 & \\-5 & ...
5
votes
1answer
452 views
Eigenvalues of certain block hermitian matrix
Suppose I have a special block, Hermitian matrix $H = \begin{bmatrix} A & B \\ B^* & A^* \end{bmatrix}$, where * denotes conjugate transpose. The blocks $A$ and $B$ are themself Hermitian in ...
5
votes
1answer
856 views
Eigenvalues of block matrices
Let $K$ be a field of characteristic 0, and consider the following block matrix
$$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
195 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 $e^{X'}$ ...
