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).
0
votes
0answers
18 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
38 views
Subset of $GL(n,R)$
I'm trying to understand why the subset of $GL(n,\mathbb{R})$ formed by the block-matrices of the following type:
$$\begin{pmatrix} A & B \\ 0 & C \end{pmatrix}$$ where
$$A \in ...
1
vote
1answer
35 views
Linear Algebra — Block Matrix Inversion
Please excuse my formatting...
$X=\left(\matrix{A & B\\C & D}\right)$ where $A,B,C,D$ are all $n\times n$ matrices. Assuming that all stated inverses exist show that
...
3
votes
1answer
156 views
Rank of a block-triagonal matrix
Given a matrix $C=\left [ \begin{matrix} A & 0 \\ B & A \end{matrix} \right ]$, where rank(A+B)=rank(B), and rank(B)>rank(A), does rank(C)=rank(A)+rank(B) hold?
A,B are Laplacian matrices.
0
votes
1answer
34 views
Singular Value Decomposition of a block diagonal matrix
For a block diagonal matrix, we have an identity for its cholesky decomposition i.e.
$chol(Z) = chol(blockdiag(A,B,...)) = blockdiag(chol(A),chol(B),...)$
(Here, $Z = blockdiag(A,B,...)$)
I want to ...
7
votes
1answer
60 views
*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$
I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form
$A\in ...
0
votes
1answer
26 views
proof about deteminant of a complex linear transformation
say I have a linear space $V$ over $\Bbb C$ and a linear transformation $T:V \to V$
such that $T=A+iB$ where $A,B \in \Bbb R^{n \times n}$
I proved already that $T_\Bbb R = \begin{pmatrix} A & -B ...
1
vote
2answers
58 views
deteminant of a block skew-symmetric matrix
If I have a matrix if the form \begin{pmatrix} A & -B \\ B & A \end{pmatrix}
how do i turn it into something like \begin{pmatrix} X & Y \\ 0 & Z \end{pmatrix}
so the determinant is ...
5
votes
5answers
136 views
How to prove that $\det(M) = (-1)^k \det(A) \det(B)?$
Let $\mathbf{A}$ and $\mathbf{B}$ be $k \times k$ matrices and $\mathbf{M}$ is the block matrix
$$\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}
...
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
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 ...
1
vote
2answers
36 views
Eigenvalue of anti triangular block matrix (skew matrix?)
I have an real anti-triangular matrix
$M=\left[
\begin{array}{cc}
A & B \\
I & 0 \\
\end{array}
\right]$
where I is an identity matrix. $A$, $B$, $I$, $0$ are all square real ...
1
vote
0answers
40 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 ...
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 ...
0
votes
1answer
27 views
Block diagonalization of a symmetric square boolean matrix
I have a symmetric square matrix with elements from $\{0,1\}$. How can I block diagonalize it only swapping lines and columns or detect it's not possible?
2
votes
3answers
181 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 ...
1
vote
1answer
72 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, ...
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
vote
1answer
88 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 ...
-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 ...
1
vote
2answers
74 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
72 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}$$
where ${I_N}$ is the identity matrix of order $N$, $M_1$ and $M_2$ are matrices of ($N\times N$) order and ...
3
votes
2answers
111 views
Eigenvalues of Block matrices with known eigenvalues
Let's have the following $(n+1) \times (n+1)$ matrix with block elements
$\mathbf{Y} = \begin{bmatrix}
\mathbf{A} & - \mathbf{w} \\
- \mathbf{w}^{T} & b
\end{bmatrix}$
where $\mathbf{A}$ is ...
1
vote
0answers
55 views
Sorting Matrix to Block structure
I have a symmetric matrix and I want it to be as block-like as possible. I don't have a clear definition. I want the smallest number of groups of non zero elements or maybe the most non-zero elements ...
3
votes
0answers
96 views
What kind of matrix/tensor notation is this?
I'm hoping someone on here recognises this and has an answer, because I'm having serious memory issues.
About a year ago, I came across the following way of representing tensors of rank $n$ in matrix ...
5
votes
3answers
335 views
Determinants of block matrices
Let $A,B \in \mathbb{R}^{n,n}$.
Now $C = \begin{pmatrix} A & iB \\ -iB & A \end{pmatrix}$ and $D = \begin{pmatrix} A & B \\ -B & A \end{pmatrix}$.
Show that $\det(C) \in \mathbb{R}$ ...
1
vote
1answer
48 views
Help regarding a weird Matrix
Hi I have a matrix of the following form arising by discretization of a system of PDEs. I am working to get the invertibility of the Matrix. Can some one help me or at least give me some reference on ...
2
votes
0answers
92 views
Properties about Matrices that can be proved by only using Block Multiplication of Matrices
I recently proved the property that product of two upper triangular matrices is an upper triangular matrices by using the block multiplication of matrices. The basic fact that was required to prove ...
0
votes
0answers
39 views
Perturbing/sensitivity of the eigenvalues of a block tridiagonal matrix
This question is related to my former question:
Checking if one "special" kind of block matrix is Hurwitz
Given the next matrix
$$
J = \begin{bmatrix}-(B+B^T) & B \\ 0 ...
1
vote
1answer
66 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
115 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 ...
0
votes
0answers
58 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 ...
