Questions tagged [matrix-theory]
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
466 questions
11
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1
answer
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Decide if a matrix is transposable
A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...
- 111
6
votes
1
answer
1k
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Largest eigenvalues of a (random) correlation matrix?
I am recently studying on eigenvalues of a (random) correltion matrix. For a $N\times N$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some ...
- 380
2
votes
2
answers
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How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?
$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists?
I am interested in this question, because my ...
- 698
3
votes
2
answers
1k
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Non-backtracking random walk in regular (finite) graphs
I know that many things are known when dealing with random walks on a finite (or even infinite) graph: mixing time, returns to origin, etc. All is based in the use of the Markovian property of such a ...
- 1,561
4
votes
0
answers
249
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Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite ...
- 2,712
7
votes
1
answer
248
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The determinant of a $4\times4$ matrix associated to some specific polynomial as follow
Let $f\in \mathbb{R}[x_1,x_2,x_3,x_4]$ defined by
$$f_a(x_1,x_2,x_3,x_4)=\prod_{1\leqslant i<j\leqslant4}(x_i-x_j)^{2a_{ij}}$$
where $a=(a_{12},a_{13},a_{14},a_{23},a_{24},a_{34})\in \mathbb{N}^6$.
...
- 1,997
5
votes
2
answers
539
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On the existence of integer square root of a $3 \times 3$ positive definite matrix
As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique.
I have read some ...
- 61
3
votes
0
answers
178
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On a matrix inequality based on the Schur-Horn theorem
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.)
Let $A_s$ denote the symmetric part of $A$...
- 2,712
12
votes
2
answers
800
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A (linear) optimization problem subject to (linear) matrix inequality constraints
Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...
- 2,712
1
vote
2
answers
1k
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A "positive diagonal plus skew-symmetric" matrix decomposition
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
My question. Do there exist an orthogonal ...
- 2,712
2
votes
0
answers
535
views
eigenvalues of a square block matrix
How can we show that there are not defective eigenvalues for this square block matrix of dimension $2d \times 2d $: \begin{bmatrix}
A&B\\-B& 0
\end{bmatrix}
where A, B are real matrices, $A =\...
- 21
2
votes
0
answers
248
views
A parametrization of stable matrices
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...
- 2,712
16
votes
7
answers
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Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the ...
- 355
1
vote
0
answers
163
views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
- 11
3
votes
0
answers
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About product of PSD matrices
In Theorem 3 in this paper, https://core.ac.uk/download/pdf/82822897.pdf, ``On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, ...
- 2,246
1
vote
0
answers
175
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Matrix Bernstein for spherical random variables
Theorem 4.1 in Tropp's Matrix Concentration Inequalities provides an exponential concentration inequality for the spectral norm of a matrix $Z = \sum_i \gamma_i B_i $, where $\gamma_i$ are an i.i.d. ...
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votes
1
answer
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Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I ...
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1
vote
0
answers
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Derivative of complex matrix pseudo inverse with respect to real and imaginary components
I have a complex non-square matrix $\mathbf{Y}\in\mathbb{C}^{n \times m}$ whose inverse I compute using the Moore-Penrose pseudo inverse, $\mathbf{Z}=\mathbf{Y^+}$.
I am interested in evaluating the ...
1
vote
0
answers
78
views
find linear approximation of non-linear matrix transform [closed]
I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...
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2
votes
0
answers
79
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Characterizing a subclass of row-orthogonal matrices
Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...
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1
vote
1
answer
103
views
On ranks of matrices with tensor structure
Fix two $2^t$ length vector of form $p=\begin{bmatrix}u_1&v_1\end{bmatrix}\otimes\dots\otimes\begin{bmatrix}u_t&v_t\end{bmatrix}$ and $r=\begin{bmatrix}w_1&z_1\end{bmatrix}\otimes\dots\...
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4
votes
0
answers
1k
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Generalizing Autonne-Takagi factorization
Autonne-Takagi factorization (Léon Autonne (1915) and Teiji Takagi (1925)) says that:
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: If $A$ is a rank-$n$ complex symmetric ...
- 10.5k
3
votes
1
answer
145
views
Connections between eigenvectors after matrix multiplication
Suppose we have an M$\times$N complex matrix $H$ and its singular value decomposition $H=U\Lambda V^*$ and an N$\times$N covariance matrix $R_s$ with its eigendecomposition $R_s = U_s\Lambda_sU_s^*$. ...
- 33
1
vote
1
answer
333
views
On a condition for a matrix sum to be zero
Let $\{Y_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices ($\mathrm{rank}(Y_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive definite ...
- 2,712
8
votes
1
answer
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Square root of a large sparse symmetric positive definite matrix
I am trying to calculate
$$Y = A^{\frac 12} X$$
where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...
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4
votes
0
answers
435
views
An upper bound on the Jordan condition number of a matrix
The Jordan condition number of a matrix $A$ is defined to be $\min_{V}\kappa(V)$, where $V$ ranges over complex matrices that satisfy $A = VJV^{-1}$ for $J$ being the unique Jordan normal form matrix ...
- 225
0
votes
0
answers
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Classifications of the indefinite generalized Cartan matrix
I want to know that the present results about classifications of generalized indefinite Cartan matrices.
I only have known that the classifications of hyperbolic matrces.
9
votes
1
answer
534
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Well known matrix inequality?
I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true?
$${...
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13
votes
3
answers
3k
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Differentiability of Eigenvalues - Perturbation Theory
first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...
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4
votes
1
answer
423
views
Is it hard to decide whether a matrix is a square of another matrix?
According to the well-know quadratic residue (QR) theory over integers, we know that it is hard to decide whether a given integer $m\in\mathbb Z_N$ is a quadratic residue (i.e., a square of another ...
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3
votes
1
answer
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views
Finding an adjacency matrix whose cube's diagonal is equal to a given vector
How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and
$$\mbox{diag}\left(A^3\right)=b$$
for some given vector $b$?
Is there a way to characterize all ...
- 503
10
votes
2
answers
5k
views
Nuclear norm as minimum of Frobenius norm product
Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix.
It is claimed that
$$
\|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
- 2,239
4
votes
1
answer
448
views
A variant of Specht's Theorem using sum of elements (rather than trace) of complex matrices?
Let us first recall Specht's Theorem. Denote by $\text{Mat}_{\mathbb{C}}(n)$ the set of all $n\times n$ matrices over the complex field $\mathbb{C}$. Let $A$ be a matrix in $\text{Mat}_{\mathbb{C}}(n)$...
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2
votes
1
answer
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An inequality regarding projection
Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b << 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define ...
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16
votes
5
answers
2k
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Expected value of determinant of simple infinite random matrix
Suppose we have a matrix $A \in \mathbb{R}^{n\times n}$ where
$$A_{ij} = \begin{cases} 1 & \text{with probability} \quad p\\ 0 &\text{with probability} \quad1-p\end{cases}$$
I would like to ...
- 355
2
votes
2
answers
565
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Regarding minimal elementary generators for $GL(n, \mathbb{Z})$
I have a result concerning the minimal number of elementary generators (and by this I mean generators which are elementary matrices) for $GL(3, \mathbb{Z})$. I'm currently working on extending the ...
3
votes
1
answer
477
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Matrix continued fractions
I am aware of the classical continued fraction in the field of real numbers, but recently I have come across the term matrix continued fraction and when I checked on the internet there are varieties ...
- 1,269
2
votes
0
answers
55
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Lower bounds on eigenvalues of Lyapunov solutions
Let $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$ and let $X\in\mathbb{R}^{n\times n}$, $X=X^\top>0$ be the solution of the following Lyapunov algebraic equation
$$
AX+XA^\top=-BB^\top....
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4
votes
1
answer
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A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
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5
votes
1
answer
8k
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Eigenvalues and eigenvectors of tridiagonal matrices
What can I say about the eigenvalues and eigenvectors of the tridiagonal matrix $T$ given as
$T = \begin{pmatrix}
a_1 & b_1 \\
c_1 & a_2 & b_2 \\
& c_2 & \ddots & \ddots \\
&...
- 73
8
votes
1
answer
2k
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Matrix elements of exponential of tridiagonal matrices
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...
- 371
0
votes
1
answer
77
views
$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$
Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m ...
- 39
2
votes
1
answer
111
views
Equality or inequality for determinant of $A_{n \times m} D_{m \times m} A^T_{m \times n}$
Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n}$. $D$ is a positive diagonal matrix and $m > n$.
Is there any equality or inequality over $|B|$, $|AA^...
- 39
3
votes
2
answers
346
views
If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal minor of order $k$, is $\frac{D_k}{D_{k-1}} > 0$ always true?
In Chapter II, Paragraph 4, Section 1 of
F. R. Gantmacher, The Theory of Matrices. Vol. 1 English translation by K. A. Hirsch. Chelsea Publishing Company. 1959. SBN 8284-0131-4,
the following ...
- 131
3
votes
2
answers
3k
views
Inverse of particular lower triangular matrix
I have an $n \times n$ lower triangular matrix $A$ where
$$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$
$$A_{ii}=1, \quad 1 \leq i \leq n,$$
and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
- 31
2
votes
1
answer
99
views
Proving symmetry of trace function of special matrix
Let $W = aI_{n\times n} + bJ_{n\times n}$, where $I$ is an identity matix, $J$ is the matrix of all ones, $a,b\in\mathbb{R}$ and a+b>0. Also, let $A = \mathbf{P} - \mathbf{p}\mathbf{p}^{T}$, where $\...
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4
votes
1
answer
289
views
A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
- 81
3
votes
0
answers
297
views
The inverse of sum of two positive matrices with almost orthogonal supports
I am interested to find an approximate formula for
$$A (A+B)^{-1} A\ ,$$
for two positive matrices $A$ and $B$ whose supports are almost orthogonal.
If the support of $A$ and $B$ are orthogonal ...
- 31
2
votes
0
answers
125
views
Ruelle-Perron-Frobenius for continuous time
I'm looking for a proof (or a reference for it) of the following result:
Let $L$ be a $n \times n$ $Q$-matrix, $p_0$ probability vector such that $L(p_0) = p_0$ and $V\colon \Omega \rightarrow \...
- 21
2
votes
1
answer
104
views
Matrix inequality with arbitrary large ratios
Let $M = (m_{ij})$ be $n \times n$ symmetric positive definite matrix. Then it can be proven that
$$ M^{1/2}A M^{1/2} \succeq M^{1/2}D M^{1/2}\succ 0
$$
so
\begin{equation}
\lambda_{\min}(M^{1/2}A M^...
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