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 ...

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0
votes
1answer
39 views

A nice proof that completely bounded (cb) norm of transpose map on $ M_n $ is n

In my research of operator algebras and their connection with machine learning I of course use the well know result: For the map $ tr:M_n \to M_n $ denoting the transpose map of matrices (meaning ...
0
votes
0answers
45 views

Image of composition of integral upper triangular matrices

For $A,B$ integral upper triangular matrices on $\mathbb{Z}^k$, do we know something about the image $\text{im}(AB)$ in terms of $\text{im}(A)$, $\text{im}(B)$, unions, intersections, determinants, ...
2
votes
1answer
90 views

Carathéodory's theorem for $SO(3)$?

Let $Q \in \operatorname{Conv} SO(3)$. Is there a way to retrieve an explicit representation of $Q$ as convex combination $Q=\sum_{k=1}^{r}{\lambda_{i}R_{i}}, R_{i} \in SO(3)$? An approximation ...
0
votes
0answers
83 views

Checking whether the determinant of a matrix is non-zero [on hold]

For a positive integer $n$, let $c$ be the number of ordered integer tripartitions $(a,b,n-a-b)$ of $n$. Let $F = ( f_{ij} )$ be a $c \times c$ matrix where $f_{ij} = (i+1)^{a + b} (i+2)^{b}$ where $i$...
0
votes
0answers
51 views

Checking whether a given matrix has a non-zero determinant

For a positive integer $n$, let $c$ be the number of ordered integers tripartitions $(a_j,b_j,c_j)$ of $n$. Now consider the $c \times c$ matrix $M$ in which the value of the $M[i,j]$ is $M[i,j]={(...
5
votes
0answers
88 views

Solving matrix equation $X^{-1}=\sum_{i=1}^n D_i X A_i$ [on hold]

Does anybody know an algorithm to solve the following matrix equation? $$X^{-1}=\sum_{i=1}^n D_i X A_i$$ where $D_i$s are diagonal and $A_i$s are symmetric matrices. It would be great to have an ...
0
votes
0answers
32 views

$nD$ rotation around a general $(n-2)$-dimensional subspace [closed]

According the Rodrigues' Rotation Formula $3D$ rotation matrix $\in$ $SO(3)$ corresponding to a rotation by an angle $\theta$ about a fixed axis specified by the unit vector $\hat{\omega}=(\omega_x,\...
-3
votes
0answers
77 views

Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [closed]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...
2
votes
0answers
55 views

How to find a closed form of following matrix's determinant [closed]

I wanna find a closed form of determinant of the following matrix $$A(n) = \begin{pmatrix} B_{1} & B_{2} & \cdots & B_{n} & 1 \\ B_{n} & B_{1} & \cdots & B_{n-1} &...
-1
votes
0answers
62 views

When will the spectral radius of a matrix reach its minimum?

Let $A$, $B$ be two $n\times n$ matrices. How to determine $s,t\in\mathbb{N}^+$ such that the spectral radius of $A^sB^t$ will reach its minimum?
4
votes
1answer
72 views

information measure for matrix that is analogous to rank

Is there a measure for matrix that is analogous to rank of the matrix, but it is continuous on matrix elements? Say, we could say the information in identity matrix $I_n$ is $n$, and when the off-...
0
votes
0answers
20 views

Number of Asymmetric, Balanced Permutation Matrices

let $\mathcal{ABP}$, the set of "asymmetric, balanced permutation matrices", be defined as the set of permutation matrices, that aren't equal their transpose but, for which the number $1$s below the ...
3
votes
0answers
25 views

Selecting columns from multiple matrices to form a well-conditioned matrix

Given multiple matrices of the same size, is there a way to select one column from each matrix to form a well-conditioned matrix? For example, given four 4-by-10 matrices A, B, C, D (real, positive, ...
0
votes
1answer
81 views

How does the rank of $C_i$ change with $i$?

Let $k$ be a field. Let $A,B\in k^{m\times n}$ and $$C_i=\pmatrix{A&B&&&\\&A&B&&\\&&\ddots&\ddots&\\&&&A&B}\in k^{im\times(i+1)n}.$$ ...
0
votes
0answers
13 views

Stucture of inverse (MP) of totally positive rectangular matrix

The special structure of inverse of non-singular totally positive square matrix (whose all entries are positive) discussed in MO(see here). The inverse has a special structure (M-matrix). With some ...
0
votes
0answers
45 views

SVD alternatives for symmetric matrices

Given any symmetric real valued matrix $A \in \mathbb{R}^{n\times n}$, I can decompose $A$ as the product of two complex matrices $$ A = E'E $$ Practically this can be done easily using SVD ...
1
vote
2answers
61 views

How can I efficiently approximate the stationary distribution of an infinite CTMC with a sparse rate matrix?

I am looking for methods to approximate the stationary distribution of an infinite CTMC with a sparse rate matrix. Each row and column of the rate matrix has a finite number of non-zero elements. ...
0
votes
0answers
34 views

Joint distribution of eigenvalue and matrix entry

Let $X=(X_{n,m})_{n,m=1}^N$ be a $N\times N$ GUE random matrix, and let $\lambda_1,\dots,\lambda_N$ denote its unordered eigenvalues. What can be said about the distribution of, say, $$\lvert\lambda_1-...
0
votes
1answer
63 views

Matrix norm inequality for C*-Algebras [closed]

Let A a $C^*$-Algebra. I have already shown that the maps $Tr, \sigma: M_n(A)\rightarrow A$ given by $Tr((a_{ij})):=\sum_{i}a_{ii}$ and $\sigma\left(\left(a_{ij}\right)\right)=\sum_{i\text{, }j}a_{ij}$...
1
vote
0answers
39 views

Matrix transformation [closed]

I want to show that $(I-H^T(-s)H(s))^{-1}$ has no poles on the imaginary axis with $H(s)=C(sI-A)^{-1}B$ and $H^T(-s)=-B^T(sI+A)^{-T}C^T$ is equivalent to $M_\gamma$ has no purely imaginary ...
6
votes
2answers
820 views

What is the Time Complexity of the Matrix Exponential?

While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called scipy.linalg.expm(). According to ...
1
vote
0answers
68 views

Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...
6
votes
1answer
212 views

A sufficient condition (or not) for positive semidefiniteness of a matrix?

Let $A\in M_{n}(\mathbb{C})$ be a Hermitian matrix. If for all $z_1,...,z_n\in\mathbb{C}$, $$\sum_{i,j=1}^{n}A_{ij}z_{i}\overline{z_{j}}\ge 0$$ then A is positive semidefinite. I do not think the ...
4
votes
1answer
93 views

Pfaffian of several skew-linear transformations / matrices

Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
6
votes
1answer
232 views

coefficient-wise powers of matrices. Reference wanted

Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
2
votes
1answer
159 views

The class of $(-1,0,1)$-matrix with all row sums and column sums equalling to $0$

Let $n$ be an even positive integer and $W_n$ be the class of all $n\times n$ matrices with entries from the set $\{-1,0,1\}$ satisfying all row sums and column sums are equal to $0$. For any $M\in ...
5
votes
2answers
291 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\...
4
votes
1answer
75 views

Perturbations on the pseudoinverse of a matrix

Given a matrix $A \in \mathbb{R}^{n\times m}$, and its perturbation $$ A_p = A + \Delta $$ is there a way to represent $$ (A_p)^{\star}= (A)^{\star} + f(\Delta) $$ where $(A_p)^{\star}$ ($(A)^{\star}...
11
votes
3answers
240 views

Product of conjugate matrices in $\mathrm{SL}(2, \mathbb{Z})$

I've read that if $M_1, \dots, M_n$ are matrices in $\mathrm{SL}(2, \mathbb{Z})$ whose product is the identity, and each is conjugate to the shear $$ \begin{pmatrix} 1 & 1 \\ 0 & 1\end{...
0
votes
1answer
86 views

Eigenvectors of symmetric positive semidefinite matrices as measurable functions

I'm currently interested in how discontinuous can get the eigenprojections of a continuous function taking values in a particular subspace of symmetric matrices. I've been searching everywhere for an ...
3
votes
0answers
85 views

Finding nearest Toeplitz matrix to a given matrix

For an arbitrary $N\times N$ Hermitian matrix $A$, I want to derive a Toeplitz matrix from $A$ such that the eigenvectors of both matrices have minimal change. Specifically I want find the Toeplitz ...
0
votes
2answers
59 views

Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix. I've seen a similar ...
4
votes
1answer
96 views

I have a very large sparse matrix, 'A', in Ax = b. What work in advance of getting 'b' can be done to reduce solving time?

This question borders between a programming and math question (more math). I have a little matrix knowledge but this is past my ability, so any help is very much appreciated. Question I have a very ...
3
votes
1answer
136 views

Is there a smooth polar decomposition for non-invertible matrices?

Every $n \times n$ real matrix $A$ has a polar decomposition $A=OP$, where $O \in O_n, P$ is symmetric positive semi-definite. $P$ is uniquely determined by $A$, by $P(A)=\sqrt{A^TA}$, and when $A$ ...
1
vote
0answers
12 views

non-symmetric weak diagonal-dominant matrix, no decoupling: (a) is positive semi-definite? (b) has dim(ker)=1?

We are considering a matrix $A=(a_{ij})_{i,j=1,\ldots,d}\in\mathbb{R}^{d\times d}$ with the following property: $a_{ii}=-\sum_{j\neq i}a_{ij}$, i.e. the matrix is not only weak diagonal-dominant, but ...
9
votes
0answers
206 views

Rationality of a certain real algebraic variety

Let $A_n$ denote the vector space of $n\times n$ antisymmetric matrices over ${\mathbb{Q}}$, where $n$ is even. Let $A_n^*\subset A_n$ denote the affine ${\mathbb{Q}}$-subvariety of invertible ...
1
vote
1answer
84 views

Does similarity imply symmetric similarity?

Let $F$ be an infinite field. If two $n×n$ matrices $A,B$ are similar in $M_n(F)$, then are they also similar via a symmetric matrix (that is, is there a symmetric matrix $Q∈GL_n(F)$ such that $A=QBQ^{...
-2
votes
1answer
106 views

Eigenvalues of cyclic tridiagonal matrix [closed]

The following matrix is the result of a special kind of balanced signed graph of order $n$. In the Matrix $n_1,n_2,..,n_k$ are positive integers, which satisfy $\sum n_i=n.$ Prove that this matrix ...
2
votes
1answer
78 views

Bounding entries of the inverse of certain zero-one matrices

It is known that the entries of the inverse of an (invertible) $n \times n$ zero-one matrix $A$ can grow exponentially in $n$. See this MO question: Bounding the absolute sum of entries of the ...
1
vote
1answer
45 views

range of singular values of sub-matrices

Assume we have a $m \times n$ matrix $A$ with real entries representing an operator $T$ on $n$ dimensional real vector space $V$. Then we select a $n-1$ dimensional subspace of $E$ of $V$ and ...
2
votes
0answers
120 views

Can the matrix exponential be equal to the elementwise exponential [closed]

Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential $e^A=\sum_{k\...
1
vote
3answers
164 views

The functional equation $T(x\otimes y)=T(x)\otimes T(y)$ on the matrix algebra

Is there a non trivial sequence $(T_{n})$ of linear operators $T_{n}$ on $M_{n}(\mathbb{C})$ such that $$T_{nm}(X\otimes Y)=T_{n}(X) \otimes T_{m}(Y) $$ where $X$ and $Y$ are in $M_{n}(\mathbb{C})$ ...
1
vote
1answer
114 views

Modified interlacing of eigenvalues

Let $A$ be a real symmetric matrix of order $n$ and $B=\begin{bmatrix}v &v &v &v\end{bmatrix}$ where $v$ is a non zero real column vector of dimension $n$. Consider $$C=\begin{bmatrix}A &...
2
votes
2answers
948 views

Encoding vectors of size $n$ in matrices which less than $2n$ rows [closed]

I have a set of vectors and each has $n$ nonnegative entries. Moreover, each entry of a vector has a quality: (1) or (2). It makes $2^n$ different possible patterns. For example, let's take two ...
1
vote
0answers
42 views

Explanation for a spectral measure [closed]

Could someone help me, please, to understand in term of entries of a Matrix $M=(m_{i,j})_{i,j\in\{1,n\}^2}$ the following measure : $$ \frac1{n} \sum_{i=1}^n \langle v_i,e_j \rangle \delta_{\lambda_i}...
3
votes
0answers
166 views

Determinant of a certain Vandermonde matrix

Is there a closed form expression for the determinant of the $n\times n$ Vandermonde-type matrix $$A = \left(\begin{array}{} 1&g_1 & x_1&g_1 x_1 & x_1^2&g_1 x_1^2 & \cdots &...
0
votes
1answer
23 views

Identifying winner of tournament (A,B testings) with none binary test result [closed]

I'm trying to setup a tournament based on votes. Let say user vote for products A, B and C. Each user is presented all possible combinations of products in random order and he picks his preferred ...
3
votes
0answers
92 views

How do I ensure that my matrix is positive definite? [closed]

I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form \begin{equation} M=\begin{pmatrix} \Sigma&P'\\ P&\Sigma \end{pmatrix} \end{equation} where $\Sigma$ is a $n\...
0
votes
0answers
43 views

How to get Wiener-Hopf decomposition of this matrix function

I met a matrix function $M(z)$ that I need to get its Wiener-Hopf decomposition that $M(z) = M_+(z) M_-(z)$ with $M_+(z)$ and $M_-(z)$ being analytic inside and outside the unit circle respectively. ...
1
vote
1answer
92 views

Find a square, stochastic matrix (w/ non-neg entries) of odd size, not a permutation matrix, with an eigenvalue other than 1 on the unit circle

...or prove that none exists. Note that such a matrix M couldn't be primitive, so there would be at least one entry equal to zero in every power M^k (Perron-Frobenius theory). Preferably the ...