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Questions tagged [matrices]

Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.

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0answers
32 views

How to minimize n-polytope's bounding box with linear transformation?

I am working on an exact algorithm for integer linear programming for my master's thesis: $Ax\leq b, x \in \mathbb{Z}^n$ $cx\rightarrow min$ For my idea to work out, I need a guarantee that n-...
4
votes
1answer
141 views

How to find the analytical representation of eigenvalues of the matrix $G$?

I have the following matrix arising when I tried to discretize the Green function, now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
0
votes
1answer
66 views

Computing spectrum of convex combination of SPD matrices given individual spectral decompositions

Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
1
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0answers
44 views

Existence of subspace which is totally non-invariant under unitary transformation

Given a complex Hilbert space $\mathcal{H}$ of dimension $d$ - interpreted as vectorspace over $\mathbb{R}$ with dimension $2d$. And the space $L(\mathcal{H},\mathbb{C}^4)$ of all linear operators ...
1
vote
1answer
104 views

How to solve this system of Matrix equations? (Coupled riccati equations?)

I am trying to solve for K in the following problem: $ 3I = A_1 + A_2 + A_3$ $ A_1 K A_1 = K_1 $ $ A_2 K A_2 = K_2 $ $ A_3 K A_3 = K_3 $ Where $I$ is the identity, $K, K_1, K_2, K_3, A_1, A_2, ...
1
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0answers
73 views

Injective differential of linear operators on a Hilbertspace

Given a complex Hilbertspace $\mathcal{H}$ of dimension $\dim(\mathcal{H}) = d$ and the set $$\mathcal{F} := \{q\in L(\mathcal{H})\vert\quad \text{rank}(q) = 4 \quad \wedge \lambda^q_{1,2} < 0\ \ \...
12
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1answer
335 views

A matrix identity related to Catalan numbers

Let $$C_n=\frac{1}{2n+1}\binom{2n+1}{n}$$ be a Catalan number. It is well-known that $$(\sum_{n\ge{0}}C_n x^n)^k=\sum_{n\ge{0}}C(n,k)x^n$$ with $$C(n,k)=\frac{k}{2n+k}\binom{2n+k}{n}.$$ It is also ...
0
votes
1answer
163 views

Proof of A Positive Definite Covariance Matrix

I would like to prove such a matrix as a positive definite one, $$ (\omega^T\Sigma\omega) \Sigma - \Sigma\omega \omega^T\Sigma $$ where $\Sigma$ is a positive definite symetric covariance matrix ...
0
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0answers
46 views

Problem of expressing a two matrices product

I think need a different method of matrix multiplication here: Suppose I have two matrices: $A = (a_{ij}) \in \mathbb{R}^{m \times m}$ and a partitioned matrix $B = (B_{ij}) \in \mathbb{R}^{mp \times ...
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1answer
100 views

What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]

Suppose we have the following symmetric matrix. $$A = \sigma^2 I + u u^T$$ What can we say about the eigendecomposition of $A$?
18
votes
1answer
1k views

A linear algebra problem in positive characteristic

Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
2
votes
2answers
136 views

Calculate percentage of symmetry of a given matrix

Is it possible to calculate a percentage that quantifies how symmetric a given matrix is? For example, even if a given matrix is not symmetric, instead of just classifying it as "not symmetric", one ...
1
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0answers
77 views

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 ...
2
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3answers
278 views

Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently? $$(D + ADA^T)^{-1}$$ Hints or references are ...
4
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1answer
121 views

Smith normal form and affine buildings

In Smith Normal Form of powers of a matrix someone has commented saying that one can reformulate many questions about Smith normal forms in the language of affine buildings. I wanted to know of a ...
1
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0answers
120 views

Sufficient conditions for all eigenvalues simple in stochastic matrix

The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple. ...
3
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1answer
221 views

Approximating the expectation of a matrix inverse

Let $$R := A \Lambda^{-1} A^H + \frac{1}{\gamma} I_n$$ where $A$ is a given $n \times m$ matrix (where $m \gg n$), $$\Lambda := \mbox{diag} \big( \lambda_1, \lambda_2, \dots, \lambda_m \big)$$ ...
0
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0answers
125 views

Closed-form solution about a matrix factorization problem?

I am doubting some equations in paper "Multi-View Learning With Incomplete Views". The paper could be found here. The problem is related to Eq.(2) and Eq.(4b) in this paper. I summarize them here (...
4
votes
1answer
144 views

Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?

I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
5
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1answer
127 views

Numerical minimization spectral norm under diagonal similarity

This question is a follow up. Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \inf_{D} \lVert D^{-1} A D\...
5
votes
2answers
254 views

Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \min_{D} \lVert D^{-1} A D\rVert_2, $$ where $D$ ...
2
votes
1answer
85 views

Symmetric orthogonal matrices with constant diagonal entries

$\newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\...
6
votes
1answer
340 views

A question on symmetric matrices

$\newcommand{\R}{\mathbb{R}}$ The question is Is there a constructive (say, parametric) description of the set (say $M_n$) of all symmetric matrices $A\in\R^{n\times n}$ such that all the diagonal ...
1
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0answers
66 views

Hubbard–Stratonovich for matrices

Can somebody share a link to papers( books) where this fornula was deduced? $e^{\frac{1}{2}\sum_{ij}K_{ij}s_is_j} =\left(\frac{\det{K}}{(2\pi)^N} \right)^{1/2} \int^{\infty}_{-\infty}...\int^{\infty}_{...
1
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0answers
124 views

Decomposition of Determinant of Sub-Matrices of a Matrix

Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means $$ {\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...
1
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0answers
76 views

Decomposition of a Matrix by Sparse Matrices

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$. $\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...
1
vote
0answers
32 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 ...
4
votes
0answers
48 views

Rank of binary matrix related to the number of positive squarefree integers less than $n$

I posted this question at the Mathematics SE, but received no response there so I am posting it here. The following fact is stated in the comments-section of sequence A013928 in the OEIS. Let $C$ ...
2
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0answers
70 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 $O$,...
1
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1answer
208 views

Dimension (manifold) of matrices with exact $r$ positive and $r$ negative eigenvalues

For the vector space $M_{n,n}(\mathbb{C})$ of $n\times n$ matrices we know that the subset $$M_{2r}:= \{A\in M_{n,n}(\mathbb{C}) \mid \mbox{rank} (A) = 2r \}$$ is a manifold of dimension $2n(2r)-(...
2
votes
1answer
131 views

When does a row standardized adjacency matrix have a real spectrum?

A colleague in spatial statistics was looking at a map with about 600 regions. For the application she's considering, the induced adjacency matrix had some undesirable properties (where two regions ...
6
votes
1answer
276 views

Is this generalization of the Hopf map for quadratic field extensions surjective?

Let $k$ be a field, and let $L$ be a quadratic extension of $k$. Denote by $\sigma$ the non-trivial element of $\operatorname{Gal}(L/k)$. Let $M_2(L)$ be the vector space over $L$ of two-by-two ...
4
votes
1answer
136 views

Positive definite matrices diagonalised by orthogonal matrices that are also involutions

Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$. I want to know when this matrix is also an involution, i.e., $P^2 = I$. If there is any ...
1
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1answer
94 views

Equivalence of matrices over a vector space

I'm trying to characterize equivalence classes of matrices over a vector space. Specifically, let $V$ be a vector space over a field $K$, let $M \in M_n(V)$ be an $n \times n$ matrix with entries in $...
5
votes
1answer
130 views

Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$ $$ A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}. $$ I'm interested in finding a ...
14
votes
2answers
504 views

What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra. A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...
0
votes
1answer
277 views

Cholesky decomposition – non-positive definite matrix

In order to pass the Cholesky decomposition, I understand the matrix must be positive definite. However, I also see that there are issues sometimes when the eigenvalues become very small but negative ...
0
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0answers
55 views

Numerical error on the spectrum of a matrix

Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small)...
4
votes
0answers
161 views

How to find eigenvalues of following block matrices?

Is there a procedure to find the eigenvalues of A? ‎ $$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^...
2
votes
0answers
44 views

Is it possible to compute a valid Laplacian matrix from an effective resistance matrix?

I am wondering whether it is possible to retrieve a node-admittance matrix $G$ (also called Laplacian matrix) in a purely resistive network composed of nets $\{1, \dots, i, \dots, j, \dots, n\}$, from ...
3
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0answers
70 views

Group generated by symmetric shears

Consider the multiplicative group generated by matrices of the form $$ \begin{bmatrix} {1} & { 0} & { c_1} & {c_3} \\ {0} & {1} & {c_3} & {c_2} \\ {0} & {0} &...
6
votes
1answer
240 views

An Optimization Problem with Complex Variables, regarding Eigenvalues of Circulant Matrices

Let $S$ be a finite subset of the complex unit circle and $1 \in S$. For each $n \in \mathbb N $, define $f_n\colon S^{n-1}\to\mathbb R$ by $$f_n(x) := \sum_{w^{n}=1}|x_1w+ x_2w^2\cdots+x_{n-1}w^{n-1}...
1
vote
1answer
85 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\...
4
votes
0answers
134 views

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 ...
11
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0answers
177 views

Fast computation of matrix product $AXA^T$ with fixed $A$?

Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
2
votes
1answer
64 views

Maximise singular value decay by sparse matrix approximation

I have a large matrix $A \in \mathbb{R}^{n \times m}$ and would like to subtract a sparse matrix $B \in \mathbb{R}^{n \times m}$ with less than $c (n+m)$ non-zero entries, where $c > 0$ is a ...
4
votes
0answers
89 views

Does this fact about the minimal polynomial give an efficient diagonalizability criterion?

I am ready to agree beforehand that this looks more like a math.SE question. I posted it there a week ago without any feedback (except for 27 views and 2 upvotes). Besides, I really need an answer. ...
15
votes
3answers
810 views

Is this lower bound for a norm of some complex matrices true?

Let $A = [a_{ij}]_{n\times n}$ be a Hermitian matrix, such that $|a_{ij}| =1$ for $i \neq j$, and $a_{ii} = 0$ for each $i$. I am interested in a tight lower bound of $\|A\|_*:=\sum_{i=1}^n |\lambda_i(...
0
votes
1answer
98 views

prove the singularity of a matrix as solution of a non-linear equation

Let $B$ ($n \times n$) and $R$ ($m \times m$) be two square matrix with $n>m>0$ who satisfie: $B=(I-KH)B(I-KH)^T+K RK^T$ with $K=BH^T(HBH^T+R)^{-1}$ and $rank(H)=m$ I would like to prove $...
1
vote
1answer
152 views

Singular values of Hadamard Product

For two Matrices $A,B \in \mathbb{R}^{m \times n}$ the Hadamard Product is defined as $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$. For a proof of convergene I require an upper (and ideally a lower) bound on ...