All Questions
Tagged with matrix-theory linear-algebra
234 questions
3
votes
1
answer
2k
views
When is the matrix norm multiplicative
Let $|| = ||_{p,q}$ be an operator norm on $\mathbb R^{n \times m}$. In General, $\|AB\|\le \|A\|\|B\|$. Is there some criterion on $A, B$ (at least for some operator norms) so that $\|AB\| = \lVert ...
- 133
11
votes
1
answer
896
views
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
2
votes
2
answers
2k
views
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
4
votes
0
answers
249
views
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
views
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
1
vote
2
answers
1k
views
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
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
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 ...
- 11
2
votes
0
answers
79
views
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 $O$,...
- 2,712
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\...
- 13.9k
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
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
9
votes
1
answer
534
views
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?
$${...
- 1,211
3
votes
1
answer
741
views
Finding an adjacency matrix whose cube's diagonal is equal to a given vector
How can I find all binary matrices $A$ such that $A^3$ is a non-negative, integer square matrix and
$$\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 $\text{Mat}_{\mathbb{C}}(n)$...
- 493
2
votes
1
answer
236
views
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 ...
- 482
2
votes
0
answers
55
views
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
$$
AX+XA^\top=-BB^\top....
- 2,712
4
votes
1
answer
414
views
A Handbook of Matrix Factorizations
I am looking for a good collection of facts regarding the various types of matrix factorizations, something like a "Handbook of Matrix Factorizations" or a very-thorough review paper. I am hoping for ...
- 143
8
votes
1
answer
2k
views
Matrix elements of exponential of tridiagonal matrices
Is there a way to compute one matrix element of the exponential of a tridiagonal matrix without having to compute the rest of the elements?
Motivation: I'm trying to find the first passage time ...
- 371
3
votes
2
answers
346
views
If $S$ is a nonsingular symmetric matrix over a number field and $D_k$ is its principal minor of order $k$, is $\frac{D_k}{D_{k-1}} > 0$ always true?
In Chapter II, Paragraph 4, Section 1 of
F. R. Gantmacher, The Theory of Matrices. Vol. 1 English translation by K. A. Hirsch. Chelsea Publishing Company. 1959. SBN 8284-0131-4,
the following ...
- 131
3
votes
2
answers
3k
views
Inverse of particular lower triangular matrix
I have an $n \times n$ lower triangular matrix $A$ where
$$A_{i,j} = {\bf x}_i{\bf x}_j^H,\quad i>j$$
$$A_{ii}=1, \quad 1 \leq i \leq n,$$
and ${\bf x}_i$ is a $1 \times k$ (row) vector, where $...
- 31
4
votes
1
answer
289
views
A property of positive matrices
Let $\mathbb{S}^{dn} \ni X \succeq 0$ with $d,n \in \mathbb{N}$, where $X \succeq 0$ indicates that $X$ is positive semidefinite. Now partition $X$ into the block form
\begin{gather}
\begin{pmatrix}
...
- 81
17
votes
2
answers
2k
views
Is the determinant the only multiplicative matrix function? [closed]
Is there a matrix invariant or property that is multiplicative, i.e.,
$$f(AB) = f(A) f(B)$$
other than the determinant? In addition, some matrix norms are submultiplicative, but is there a ...
- 1,465
2
votes
0
answers
122
views
Number of distinct rows and columns in a matrix with bounded number of entries
How many distinct rows and columns a real square matrix can have (at least in symmetric case) such that rank of matrix is $r$ and entries:
are from $\{-b,-b+1,\dots,0,\dots,b-1,b\}$?
are from $\{-b,-...
- 13.9k
1
vote
0
answers
172
views
A vanishing sum of symmetric matrices
Let $\{G_i\}_{i=1}^N\in\mathbb{R}^{n\times m}$ be a set of full column rank matrices (i.e., $\mathrm{rank}(G_i)=m$ for all $i$) and $\{P_i\}_{i=1}^N\in\mathbb{R}^{m\times m}$ be a set of positive ...
- 2,712
6
votes
0
answers
99
views
What is this matrix decomposition called and does it exist always? - II
Given a rank $2r$ matrix $M\in\Bbb Q_{\geq0}^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank at most $r$ such that $M=M_+-...
- 13.9k
0
votes
1
answer
138
views
On sum of matrices
Suppose we have a matrix $M\in\Bbb Q^{n\times n}$ with no $0$ elements and we write as sum of two matrices $M_1$ and $M_2$ on following constraint.
$M_{1,ij}=M_{ij}$ and $M_{2,ij}=0$ or $M_{1,ij}=0$ ...
- 13.9k
4
votes
1
answer
294
views
What is this matrix decomposition called and does it exist always?
Given a rank $2r$ matrix $M\in\Bbb Q^{n\times n}$ can we find two matrices $M_+\in\Bbb Q_{\geq0}^{n\times n}$ and $M_-\in\Bbb Q_{\geq0}^{n\times n}$ each of rank $r$ such that $M=M_+-M_-$ holds?
...
- 13.9k
1
vote
1
answer
192
views
Is there any relation between weights in the eigenvector (corresponding to least eigenvalue) and the columns of a correlation matrix?
This problem is motivated from one of my pattern mining research projects. Any helpful suggestions will be highly appreciated.
Consider an $n \times n$ correlation matrix A such that all the off-...
- 141
0
votes
1
answer
111
views
A conjugation matrix $X\in \mathbb{C}^{ n\times p}$ where $p< n$
Given a Hermitian positive definite matrix $A\in \mathbb{C}^{n \times n}$ and a Hermitian matrix $B\in\mathbb{C}^{ p\times p},$ find the matrix $X$ so that $X^HAX=B$ holds where $X^H$ denotes ...
- 21
-1
votes
1
answer
195
views
Determinant of $Z^TZ$ [closed]
If one is looking at the characteristic polynomial of the $m \times m$ dimensional matrix $Z^TZ$ then apparently the coefficient of $(-1)^{m-k}$ in it can be written as, $\sum_{U \subset [m], V \...
- 2,246
10
votes
1
answer
615
views
A curious determinantal inequality I
Let $A, B$ be Hermitian matrices. Does the following hold?
$$\det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)$$
As usual, $|X|=(X^*X)^{1/2}$. Clearly, quantities on both sides are no less ...
- 1,748
6
votes
2
answers
1k
views
Nontrivial lower bound on the sum of matrix norms
Let $X, V\in\mathbb{R}^{n\times r}$ such that $X^\top V$ is symmetric. The central quantity I care about is
\begin{equation}
\|XV^\top\|_{F}^2+\|X^\top V\|_{F}^2 +[\text{Tr}(X^\top V)]^2.
\end{...
- 515
-1
votes
1
answer
305
views
A simple matrix multiplication query [closed]
The entries of $\begin{bmatrix}a&b\\c&d\end{bmatrix}\begin{bmatrix}a'&b'\\c'&d'\end{bmatrix}=\begin{bmatrix}aa'+bc'&ab'+bd'\\ca'+dc'&cb'+dd'\end{bmatrix}$ are curiously given ...
- 13.9k
6
votes
1
answer
777
views
Is every real matrix conjugate to a semi antisymmetric matrix?
Is it true to say that every matrix $A\in M_n(\mathbb{R})$ is similar (conjugate) to a matrix $B=(b_{ij})$ with $b_{ij}=-b_{ji}$ for all $i\neq j$?(With some abuse of terminology,a matrix $B$ with ...
- 356
3
votes
1
answer
416
views
What's the best orthonormal matrix to align two matrices in the operator norm sense?
Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that
\begin{equation}
UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F},
\end{equation}
where $USV^\top$ is ...
- 515
1
vote
2
answers
485
views
Closed form for integral of function of a symmetric positive definite matrix
Let $M$ be a real symmetric positive definite matrix of size $n \times n$, and let $\log M$ denote its (principal) matrix logarithm.
Is it possible to evaluate the following integral in closed form?
...
- 1,538
1
vote
1
answer
129
views
Redistribute diagonal entries of a matrix
Let $d = (d_1,...,d_k)^t$ with positive entries. Denote $D:=diag(d)$ and let $m > k$. What are sufficient conditions on $d$ and $m$ so that there exists $V \in \mathbb{R}^{m \times k}$ with:
$V$ ...
- 185
1
vote
0
answers
122
views
Fullrankness of sum of time shifts
I am working with finite Gabor frames and in this context a problem appeared which I am trying to solve for a couple of weeks now.
Given a $(p,k,1)$ cyclic difference set for $\mathbb{Z}_p$ which is ...
- 93
1
vote
0
answers
188
views
Maximum singular value of sum of an Hermitian and an anti-Hermitian matrix
Let $H$ be an $n\times n$ Hermitian matrix and $A$ an $n\times n$ anti-Hermitian matrix, i.e. $H^\dagger = H$, $A^\dagger = -A$. Consider their sum $S= H+A$. Let $\{\sigma_i(S)\}_{i=1,\dots,r}$ denote ...
- 11
2
votes
2
answers
123
views
Behavior of matrix rank under thresholding of its elements
Let $A$ be a $n \times n$ real matrix of, say, rank $r$. Consider the matrix $$\max \{0,A\}$$ whereby each negative element of $A$ is set to $0$ and the non-negative elements are left unchanged. Is ...
- 2,246
3
votes
0
answers
1k
views
Eigenvalues of block-hermitian matrices with zero diagonal blocks
I have a matrix of the form
$$D = \left( \begin{array}{cc} 0 & C \\ C^{\dagger} & 0 \end{array} \right)$$
where $C$ is not necessarily hermitian. In general, can we say anything about the ...
- 47
2
votes
0
answers
167
views
Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices
Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$.
Let the singular value decomposition of $A$ be $A=U \Sigma V^...
- 323
7
votes
1
answer
3k
views
When does the determinant distribute over addition?
When does $\det(A+B)=\det(A)+\det(B)$ hold?
I actually wonder if there is an easy answer for when $Per(A+B)=Per(A)+Per(B)$.
- 13.9k
17
votes
3
answers
2k
views
Finding the nearest matrix with real eigenvalues
In this thread on MATLAB Central, I found a discussion on finding the nearest matrix with real eigenvalues. The first hypothesis was to simply truncate the complex part of the eigenvalues. So, given ...
- 185
6
votes
2
answers
254
views
A correlation matrix problem
I have a linear algebraic question about an arbitrary correlation matrix. If
$$
\lambda_{\min}
\begin{pmatrix}
1 & \rho_1 & \rho_2
\\
\rho_1 & 1 & \rho_3
\\
\rho_2 & \rho_3 & ...
- 719
3
votes
0
answers
56
views
Equivalence Classes of a Subgroup of Similarity Transformations
Let $X$ be a real, finite-dimensional vector space and $A, B, C,$ and $D$ be matrices on $X$. I'm interested in the similarity classes of the block matrices
$$
\begin{bmatrix}
A & B\\
C & D\\
...
- 263