All Questions
Tagged with matrices matrix-theory
248 questions
32
votes
0
answers
649
views
Existence of orthogonal basis of symmetric $n\times n$ matrices, where each matrix is unitary?
For a positive integer $n$, let $S_n$ denote the set of $n\times n$ symmetric matrices over $\mathbb{C}$. As a complex vector space, this set has dimension $\mathrm{dim}(S_n)=\binom{n+1}{2}$. The ...
- 481
23
votes
0
answers
8k
views
An $n \times n$ matrix $A$ is similar to its transpose $A^{\top}$: elementary proof?
A famous result in linear algebra is the following.
An $n \times n$ matrix $A$ over a field $\mathbb{F}$ is similar to its transpose $A^T$.
I know one proof using the Smith Normal Form (SNF). ...
- 3,320
22
votes
2
answers
14k
views
Infinite matrices and the concept of "determinant"
Suppose we have an infinite matrix A = (aij) (i, j positive integers). What is the "right" definition of determinant of such a matrix? (Or does such a notion even exist?) Of course, I don't ...
- 1,861
21
votes
3
answers
51k
views
What is the time complexity of truncated SVD?
Full SVD, on an $m \times n$ matrix $A$, [U,S,V] = svd(A), would cost $O(m^2n + mn^2 + n^3)$ time. But what is the time complexity if we only need the $k$ largest ...
- 327
20
votes
1
answer
25k
views
When does the $4\times 4$ 'false Sarrus rule' compute the determinant correctly?
This question is most probably not research level, but I thought that the MO folks might like it... Feel free to close.
Here is the motivation: If you have ever taught a maths course for engineers ...
- 12.7k
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
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
17
votes
1
answer
3k
views
2x2 subdeterminants of a matrix
If N>2, it is well known that if two invertible NxN matrices A and B have the same determinants of any 2x2 corresponding submatrices, then A=B or A=-B.
Given then all these 2x2 determinants of an ...
- 1,309
16
votes
7
answers
2k
views
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
16
votes
2
answers
504
views
The number of 0-1 normal matrices
Let $A\in\{0,1\}^{n\times n}$ be a $n\times n $ matrix with entries in the discrete set $\{0,1\}$.
My question: What is the number of matrices in $\{0,1\}^{n\times n}$ that are normal, that is, ...
- 2,712
15
votes
1
answer
782
views
Is SL(n,Z[x]) generated by transvections?
Is $\mathrm{SL}(n,\mathbb{Z}[x])$ equal to $E(n,\mathbb{Z}[x])$, the subgroup generated by transvections?
- 281
14
votes
1
answer
751
views
Is this "semi-tensor product" something recently invented? Are there other usages of it?
The context: I was reading a paper in which they used the following definition called "Semi-Tensor Product" (STP) or "Cheng" product (In honor to its "inventor" D. Cheng):...
- 344
13
votes
2
answers
6k
views
Parametrization of positive semidefinite matrices
We know that a real, symmetric, positive definite matrix $A$ of size $n\times n$ can be parametrized by a vector $\theta$ of $\frac{n(n+1)}{2}$ parameters thanks to the Cholesky decomposition:
$$
A = ...
- 313
13
votes
2
answers
414
views
Is every finite-order unimodular matrix conjugate to a $0,1,-1$ matrix?
Problem. Given a matrix $A\in\mathrm{GL}(n,\mathbb{Z})$ such that $A^k=1$ for some $k\geq 1$, is there a matrix $g\in\mathrm{GL}(n,\mathbb{Z})$ such that $gAg^{-1}$ has only $0$, $1$, and $-1$ as ...
- 23.3k
13
votes
1
answer
275
views
What is the symmetry group fixing norms of elements of a unitary matrix?
Let $N\geq1$ be an integer and let us say that two matrices $U,V\in U(N)$ are related if $|U_{ij}|=|V_{ij}|$ for all indices $1\leq i,j\leq N$.
When exactly are two unitary matrices related in this ...
- 8,086
12
votes
2
answers
800
views
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
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
11
votes
1
answer
453
views
A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
10
votes
2
answers
1k
views
When the sum of positive definite matrices converges, does the sum of the norm of the associate matrices converges?
Suppose $A_k>0$ (which means they are positive definitive square $n\times n$-matrices with $n>1$).
If $\sum_{k=1}^\infty A_k$ exists, then
$\sum_{k=1}^\infty \|A_k\| < +\infty$,
Where $\|A\|=\...
- 173
10
votes
1
answer
600
views
is it possible to have two non-isomorphic non-regular graphs with the same adjacent spectrum and the same laplacian spectrum?
For two regular graphs $G$ and $H$, it is possible for them to share the same adjacent spectrum and the same laplacian spectrum. While, on the other hand, is it possible to have two non-regular graphs ...
- 103
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|)$$
As usual, $|X|=(X^*X)^{1/2}$. Clearly, quantities on both sides are no less ...
- 1,748
10
votes
2
answers
5k
views
Nuclear norm as minimum of Frobenius norm product
Nuclear, or trace, or Ky Fan, norm of a matrix is defined as the sum of the singular values of the matrix.
It is claimed that
$$
\|X\|_\sigma = \min_{UV^T=X} \|U\|\|V\| = \min_{UV^T=X} \frac{1}{2}(\|...
- 2,239
10
votes
0
answers
230
views
Matrix identities in two variables
The famous Amitsur-Levitzki Theorem states that the algebra $M_n(\mathbb C)$ satisfies no polynomial identity of degree less than $2n$ and it satisfies
$$p(x_1,\dots,x_{2n}) = \sum_{\sigma \in \frak ...
- 25.5k
9
votes
3
answers
544
views
Product of a Finite Number of Matrices Related to Roots of Unity
Does anyone have an idea how to prove the following identity?
$$
\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix}
x^{-2j} & -x^{2j+1} \\
1 & 0
\end{pmatrix}\right)=
\begin{cases}
...
- 309
9
votes
1
answer
253
views
Linear subspaces of $\mathrm{GL}_n(\mathbb{R})$ whose inverses are also linear subspaces
$\DeclareMathOperator\GL{GL}$We will call a subset $S \subset \GL_n(\mathbb{R})$ a linear subspace if it is of the form $S = S'\cap \GL_n(\mathbb{R})$ for some $S'\subset M_n(\mathbb{R})$ which is a ...
- 2,178
9
votes
2
answers
1k
views
On closest unitary matrix
In this question $\|A\|_p$ is the normalized $p$-th Schatten norm which is defined to be $\left(\mathbb E_{i} \lambda_i^p\right)^{1/p}$, where $\lambda_i$ are singular values of matrix $A$.
Suppose ...
- 901
9
votes
3
answers
389
views
Is there a standard name for the following type of linear operator?
Is there a standard name for a linear operator $T$ on a finite dimensional vector space satisfying $T^n=T^{n+1}$ for some $n\geq 1$ or, equivalently, $T$ is a similar to a direct sum of a nilpotent ...
- 38.6k
8
votes
7
answers
1k
views
One observation of special type of square matrix exponentiation
I was studying the following type of matrices,
$$
A = \begin{pmatrix}
1 & x_{12} & \cdots &x_{1n}\\
0 & x_{22} & \cdots &x_{2n}\\
\vdots\\
0&\cdots&0&x_{nn}
\end{...
- 271
8
votes
1
answer
1k
views
Closed form solution for $XAX^{T}=B$
Let $d \times d$ matrices $A, B$ be positive definite. Is there a closed form solution for the following quadratic equation in $X$?
$$X A X^{T} = B$$
Thank you.
- 329
8
votes
2
answers
519
views
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
- 52.3k
8
votes
2
answers
12k
views
Relation between eigenvalues of $A$ and $A^TA$?
For an $n\times n$ diagonizable matrix $A$, is there a relation between the eigenvalues of $A$ and the eigenvalues of $A^TA$?
I ask this because I am looking into the relation between $A$ and $A+cI$, ...
- 181
8
votes
3
answers
663
views
Representation theorem for matrices (reference request)
Motivation. If $A \in \mathbb{C}^{n \times n}$ is self-adjoint (or, more generally, normal), then we all know that
$$
A = \sum_{k=1}^n \lambda_k \, h_k \otimes h_k,
$$
where $\lambda_1,\dots,\lambda_n$...
- 12.5k
8
votes
1
answer
286
views
Cardinality of the maximum points of the determinant on matrices with entries in [-1, 1]
By multilinearity, the maximum of the determinant on matrices with entries in the interval [-1, 1] is attained at a {-1, 1}-matrix. By the following example, the maximum is attained at infinitely many ...
- 745
8
votes
1
answer
5k
views
Eigenvectors of Kronecker Product [closed]
Conjecture If $A$ and $B$ are two complex square matrices, then every eigenvector of $A\otimes B$ is of the form $x\otimes y$, where $x$ is an eigenvector of $A$ and $y$ is an eigenvector of $B$.
...
8
votes
1
answer
2k
views
"Additive version" of Kronecker product
Let $A$ and $B$ be two square matrices with complex entries.
Let $\lambda_1, \ldots, ,\lambda_n$ be the Eigenvalues of $A$ and
$\mu_1, \ldots, ,\mu_m$ be the Eigenvalues of $B$.
Then the Eigenvalues ...
- 3,031
8
votes
1
answer
356
views
Rank of a combinatorial matrix
For any $2n$ with $n\in\mathbb{N}$, we conjecture the matrix $A\in\mathcal{M}_{3n\times 2n}$ to have rank $\lceil{\frac{5n-1}{3}}\rceil$ where $$A=\left(\begin{array}{cccccc}
0 & & & &...
- 83
8
votes
1
answer
340
views
General Sylvester's linear matrix equation
For what conditions on $A$, $B$ and $C$ (square matrices of size $n$) would there be a unique solution to
$$
ABX + AXC + XBC = D,
$$
for any $D$? Can one expect a characterization similar to the ...
- 1,216
8
votes
1
answer
490
views
Determinants (and traces) of linear maps of matrices
Let $k$ be a field or a commutative ring with unit and let $F:M_n(k)\to M_n(k)$ be a $k$-linear map. Suppose that $F$ is given in the form $F(X) = A_1XB_1 + \cdots + A_m X B_m$ for some $A_i,B_i\in ...
- 7,127
8
votes
1
answer
1k
views
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 ...
- 83
7
votes
1
answer
510
views
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$
where $\lVert \rVert$ is the ...
- 1,043
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
7
votes
1
answer
3k
views
What are known properties of matrices where off-diagonal elements are 1?
Consider a matrix where the diagonal entries are anything but the off-diagonal entries are all one. I was able to find a formula for the determinant of this matrix, but what are other known properties?...
- 277
7
votes
2
answers
1k
views
Matrix groups and presentation
Suppose $K$ is a number field and I have a subgroup of $\operatorname{GL}_2(K)$ for which I know a (finite) set of generators. Is there an algorithm that gives me a presentation of the group?
More ...
- 1,271
7
votes
1
answer
388
views
Questions on symmetric Hadamard matrices
Definitions: An $n\times n$ Hadamard matrix (HM for short) is a matrix whose entries are either $1$ or $−1$ and whose rows are mutually orthogonal.
If $A$ is a symmetric matrix, then $A = A^T$ and if $...
- 696
7
votes
1
answer
244
views
Generators for the semigroup $\mathrm{SL}(n,\mathbb{N})$
For $2\times 2$ matrices we have the following result.
Any matrix in $\mathrm{SL}(2,\mathbb{Z})$ with nonnegative entries can be obtained from $\mathrm{Id}_2$ by repeatedly adding one column to ...
- 301
7
votes
0
answers
131
views
Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 3,425
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
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
6
votes
2
answers
821
views
Explicit solution to a Rayleigh quotient equation
For 5 months! I have been struggling to solve the following equations analytically without numeric method (ie, Newton method):
Main equation:
$$
\biggl(M^2-\cfrac{\mathbf{x^{\text{T}}}M^2\...
- 163
6
votes
1
answer
585
views
On a trace condition for positive definite $2\times 2$ block matrices
Consider the following block matrix
$$
X = \begin{bmatrix} A & C \\ C^\top & B\end{bmatrix},
$$
where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{m\times m}$, and $C\in\mathbb{R}^{n\times m}...
- 2,712