All Questions
Tagged with matrix-theory linear-algebra
234 questions
2
votes
2
answers
132
views
Invertibility of one matrix constructed by order n subgroup of symmetric group
Let $S_n$ be the symmetric group on $n$ elements $\{ 1,2,\dotsc,n \}$ and $G$ be a subgroup of $S_n$ of order $n$. Denote the elements in $G$ by $\{ \sigma_1,\dotsc,\sigma_n \}$. Let the matrix $A=(\...
- 21
2
votes
0
answers
104
views
Find an $a \times m$ submatrix of an $n \times m$ matrix with smallest rank
Given a matrix $n \times m$, I want to find the submatrices $a \times m$ by selecting $a$ columns such that their rank is minimal. Can this problem be solved efficiently?
- 1,207
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 27
2
votes
0
answers
79
views
Does every $(n-1)^2 + 1$-dimensional subspace of $n\times n$ Hermitian matrices that contains identity, contain a rank-1 matrix?
Let $M_i$, $i=1,\dots,(n-1)^2+1$, $M_1 = 1_{n\times n}$ be a set of linearly-independent Hermitian $n\times n$ matrices. Show that there exists a rank-1 matrix $P$, which is a linear combination of $...
- 628
8
votes
4
answers
379
views
Traceless Hermitian matrices with simultaneously vanishing Rayleigh quotients
Let $D$ be an integer greater than 1. What is the largest number $N$, such that for all sets of $N$ Hermitian $D\times D$ traceless matrices $M_i$, $i=1,\dots,N$, there exists a non-zero complex ...
- 628
2
votes
0
answers
75
views
Smallest dimension, on which a set of matrices acts non-trivially
Let $A_i$, $i=1,\dots,N$, be a finite set of $D<\infty$ dimensional Hermitian matrices. Let $d$ be the smallest number for which there exists a unitary $D$-dimensional matrix $U$, and Hermitian $d$-...
- 628
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
6
votes
0
answers
111
views
Factorization to sparse matrices
$\newcommand{\lrank}{\operatorname{lrank}}$
$\newcommand{\rank}{\operatorname{rank}}$
Given a matrix $A$, we can define its Hamming weight, $w(A)$, as the number of non-zero elements in it.
Now, given ...
- 3,319
1
vote
0
answers
45
views
Rank of Hadamard product of column-wise polynomial evaluations and row-wise exponential evaluations
Consider the Hadamard product $A \odot B$ between two special matrices $A,B \in \mathbb{R}^{n \times m}$. The columns of $A$ are evaluations of polynomials, while the rows of $B$ are evaluations of ...
- 21
0
votes
0
answers
32
views
Eliminating nullity for enhanced non-singularity
If we have an
$n\times n$ matrix $A$ with entries either $0$ or $1$, where all diagonal entries are $0$ and the rank is $k<n$, can we reach full rank by changing exactly $n-k$ zero off-diagonal ...
- 4,058
1
vote
0
answers
255
views
Interpreting positive semidefinite matrix as a graph
Given any symmetric matrix $S \in \mathbb{R}^{n \times n}$, if $S \succeq 0$, is there a way to encode $S$ into a graph such that it takes into account the positive semidefinite constraint, and ...
- 275
2
votes
1
answer
264
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
- 637
2
votes
0
answers
137
views
Decompose a rational matrix as an integer matrix and an inverse of integer matrix
Suppose we have a non-singular rational matrix $Q$, consider the the $\mathbb{Z}$-span of the columns of $Q$ and $Q^{-1}$, denote it as $H = {\rm Span}_{\mathbb Z} \{ Q(\mathbb Z^{n}), Q^{-1}(\...
- 823
1
vote
2
answers
137
views
Methods to solve for a matrix whose entries satisfy certain properties
(This question is a repost of a deleted question I asked, because the previous version had several elements missing)
Setting
For fixed $N \in \mathbb{N}$, I wish to compute the entries of a matrix $...
- 135
1
vote
0
answers
155
views
Some kind of product of two 2d tensors to create a 3d tensor?
I recently need to apply the following concept of product of two 2d tensors to create a 3d tensor (tensors understood as generalized arrays):
given two 2d tensors $A_{m\times n}$ and $B_{n\times p}$, ...
- 461
1
vote
1
answer
141
views
Minimal number of linearly dependent rank-1 projectors
What is the minimal number of linearly dependent rank-1 projectors $\vec v \vec v^t$ in dimension n, under the condition that every set of n column vectors $\vec v$ is linearly independent.
PS: the ...
- 1,207
0
votes
0
answers
99
views
Efficient method to determine minimum eigenvalue of $2 \times 2$ block diagonal matrix
Suppose $H$ is a $2 \times 2$ block-diagonal symmetric matrix in $\mathbb{R}^{2^N \times 2^N} $. That is
$$ H = \begin{pmatrix} A_1 & 0 & \cdots & 0\\ 0 & A_2 & \cdots & 0 \\
...
- 131
5
votes
1
answer
510
views
A potential new norm for matrices and Horn's inequalities
I am investigating a function defined in terms of the singular values of matrices. Initially, I simplified the problem by focusing on the eigenvalues of $2 \times 2$ Hermitian, positive-definite ...
4
votes
0
answers
98
views
A question on products of linear combinations of complex matrices
Suppose that $A_1, \dots , A_n, B_1, \dots , B_n \in \Bbb C^{d \times d}$ and that, for every $x \in \mathbb{C}^n$, the following holds
$$\left( \sum_i x_i A_i \right) \left( \sum_i x_i A_i \right)^{...
- 69
0
votes
1
answer
224
views
Faulty algorithm for simultaneous diagonalization?
I found a simple algorithm for simultaneous diagonalization of two commuting matrices (Nordgren - Simultaneous Diagonalization and SVD of Commuting Matrices), which seemed to be well-founded. For ...
- 103
3
votes
1
answer
144
views
On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
3
votes
1
answer
181
views
The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
- 4,058
1
vote
0
answers
72
views
Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
- 11
1
vote
1
answer
70
views
A question about the sign of quadratic forms on nonnegative vectors
Let $M$ be a real square matrix of order $n\ge 3$.
Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$.
Can ...
- 13
2
votes
0
answers
53
views
Robustness of largest singular vectors with respect to noise
I would like to find a result that shows that the largest right-singular vectors of a data matrix are in some sense robust with respect to low-variance noise perturbations. Specifically, let $X = U D ...
- 61
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
0
votes
0
answers
168
views
Multiplying by Loewner-ordered matrices
Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
- 93
2
votes
1
answer
317
views
Classification of congruent integer matrices
I am interested in the following question:
Let $A,B\in\text{Mat}(2n\times2n;\mathbb{Z})$ be two integer matrices with the property that $\text{det}(A-A^T)=1=\text{det}(B-B^T)$. Are there known ...
1
vote
0
answers
42
views
What is the complexity of the matrix multiplication closure for a given generating system?
Given a generating set of $k$ matrices $X = \{M_1, M_2, \ldots, M_k\}$, with $M_i\in \mathrm{Mat}(\mathbb{C},n)$, what is the worst case complexity for computing the algebraic closure w.r.t. matrix ...
- 21
1
vote
0
answers
146
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 573
0
votes
0
answers
92
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
2
votes
1
answer
141
views
On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
- 4,058
4
votes
1
answer
259
views
Why do these polynomials split almost in the middle?
Start with a palindromic sequence of integers $(a_0, a_1, \ldots, a_{n+1})$, i.e. $a_j=a_{n+1-j}$, and put $a_j:=0$ for $j<0$ and $j>n+1$. You may readily guess that the choice of the binomial ...
- 13.4k
1
vote
0
answers
109
views
Relation between the dimension of vector spaces and dimension of the space [closed]
Let $A \in \mathrm{GL}(d, \mathbb{R})$ be an irreducible matrix. Assume that $\{V_{n}\}_{n\in \mathbb{N}}$ is a non-zero proper subspace $\mathbb{R}^d$ with dimension $t<d,$ such that $AV_{n}=V_{n+...
- 133
8
votes
2
answers
414
views
Recovering eigenvalues of a matrix from its $p$th compound matrix
This question was motivated by Find the determinant of a matrix given the determinant of all $p\times p$ sub-matrices?. Let $A$ be an $n\times n$ matrix over a field. Suppose
we are given the $p$th ...
- 50.8k
2
votes
2
answers
104
views
Inequality for matrix with rows summing to 1
Let $A$ be real matrix with $M > 1$ rows and $K > 2$ columns, and each entry $a_{m,k} \in (0,1)$, with each row summing to $1$. For all $m$
$$
\sum_{k=1}^{K} a_{m,k} = 1
$$
I want to find out if ...
- 23
4
votes
1
answer
720
views
Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
- 287
0
votes
1
answer
125
views
Special type of normal form of matrix in principal ideal domain
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}$I want to ask the following, Given $X \in n \times n$ matrix that all the elements are integers and $X=X^{T}$ is symmetric.
Can one always ...
- 145
2
votes
0
answers
506
views
Finding a basis for the range of a linear function
I realize this question is not high level but I have posted it on Math Stackexchange:
Stackexchange question
and have received some upvotes but no answers or comments, so I am trying here.
I will need ...
- 121
2
votes
0
answers
113
views
Product of two involutions in $\mathrm{PSL}_2(D)$
Let $D$ be a division ring and $\mathrm{PSL}_2(D)$. Suppose that $\overline{A}\in\mathrm{PSL}_2(D)$ where $A\in \mathrm{SL}_2(D)$. If $\overline{A}$ is identity, then $\overline{A}$ can express two ...
- 141
1
vote
0
answers
69
views
Does there exist a canonical form for normal matrices which extends the following embedding?
Given an unordered pair of complex numbers $\{w,z\}$, we can associate to it the complex matrix
$$\frac 1 2\left[\begin{matrix}w + z + \frac{\left(w - z\right)^{2} + \left|{w - z}\right|^{2}}{2 \left|{...
- 4,943
7
votes
1
answer
511
views
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$
where $\lVert \rVert$ is the ...
- 1,043
0
votes
0
answers
150
views
Sequence in matrices converging to Identity
Let $M_{k}(\mathbb{C})$ be the set of $k \times k$ complex matrices. I am trying to find a sequence of polynomials $P_{n}: M_{k}(\mathbb{C}) \to M_{k}(\mathbb{C})$ (or continuous functions $f_{n}$) ...
- 31
3
votes
1
answer
102
views
Multiplicative identity of determinant of multiplicative action of a polynomial on a quotient ring (companion matrices)
Let $A$ be a commutative ring with $f,g\in A[x]$ monics. Consider the $A$-linear endomorphism $\mu_g^{(f)}\in \mathrm{End}_A\tfrac{A[x]}{\langle f\rangle}$ given by multiplication by $g$.
For monics $...
- 10.5k
3
votes
1
answer
399
views
Maximal common isotropic subspace for a finite family of skewforms
Let $V$ be a vector space of dimension $n$ over a field $F$. An alternating bilinear form $\alpha\colon V \times V \rightarrow F $ will be called a skewform. A subspace $W$ is isotropic for $\alpha$...
- 923
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
3
votes
1
answer
2k
views
Eigenvalues of a block matrix with zero diagonal blocks
Suppose $A$ is a $k_1\times k_2$ matrix with real entries, $k_1<k_2$. Let $M$ be the matrix
\begin{equation}
M:=\begin{pmatrix}
0_{k_1} & A\\ A^\top & 0_{k_2}
\end{pmatrix},
\end{equation}
...
- 53
3
votes
1
answer
740
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 55
2
votes
0
answers
176
views
System of matrix equations
Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$
Question: Is ...
- 245
2
votes
0
answers
99
views
When does a matrix subspace contain a full rank matrix?
Cross-posted at Math SE
Let $S\subseteq M_{n,m}(\mathbb{C})$ be a $d$-dimensional subspace of the space of $n\times m$ complex matrices (with $n\leq m$, say). I am interested in figuring out ...
- 131