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6 votes
3 answers
256 views

Determine unknown matrix function of particular form from known points

I encountered the following problem recently in a practical context. Fix $n \ge 1$. Suppose $f$ is an unknown function $\mathbb C ^ {n \times n} \to \mathbb C ^ {n \times n}$ of the form $$ X \mapsto ...
0 votes
0 answers
131 views

On a matrix equation with Kronecker product

Is there any work on the matrix equation in unknowns $X, Y \in {\Bbb C}^{n \times n}$ $$(X \otimes Y + Y \otimes X) \operatorname{vec}(A)=0$$ where $\otimes$ is the Kronecker product? Or, in general, ...
3 votes
0 answers
142 views

Solvability of a matrix exponential equation - generalized matrix logarithm

For a given invertible real matrix $G\in \mathrm{GL}_d$ with $\det G>0$, we ask for a solution $B$ of the matrix exponential equation $$ G = \exp(B) \exp\bigl(\tfrac{1}{2}(B^T-B)\bigr) . $$ Basic ...
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{...
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.
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 ...
3 votes
2 answers
354 views

Solving linear matrix equation

Given matrices $A, B, C' \in \Bbb R^{2 \times 6}$, where $'$ denotes matrix transposition, and matrix $L \in \Bbb R^{2 \times 2}$, how can one solve the following linear matrix equation in $X \in \Bbb ...
1 vote
0 answers
84 views

In matrix product, differentiate one element with respect to another element

Background Consider a system (roughly) along the lines of those shown in Sims, C. A. (2002). Solving linear rational expectations models, where you have $$ AX_{t+1} = CX_t + M $$ where matrix $M$ is a ...
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 ...
4 votes
3 answers
239 views

Properties of matrix $X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}$

Let $\{\alpha_i\}_{i=1}^n$ be complex numbers such that $|\alpha_i|<1$, and consider the following $n\times n$ structured matrix $$ X=\left[\frac{1}{1-\bar\alpha_i \alpha_j}\right]_{ij}. $$ Such ...
1 vote
0 answers
101 views

Controlling the rank of a Matrix product

Let $\bf{Q}$ be an $(m-k)\times m$ matrix satisfying $\bf{QQ}^H=\bf{I}$, $\bf{W}$ an $m\times m\ell$ matrix of the form $$\bf{}W=\left[\begin{array}{ccccc}{\bf{w}}_1 &\bf{0} &\bf{0}&\...
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, ...
5 votes
2 answers
539 views

On the existence of integer square root of a $3 \times 3$ positive definite matrix

As far as I know, a real square matrix $M$ has a real square root if $M$ is positive semidefinite, i.e., if all eigenvalues are nonnegative. And, in fact, its square root is unique. I have read some ...
3 votes
0 answers
178 views

On a matrix inequality based on the Schur-Horn theorem

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and (strictly) positive eigenvalues. (Notice that $A$ is not required to be symmetric.) Let $A_s$ denote the symmetric part of $A$...
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 ...
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 ...
3 votes
1 answer
1k views

Completing the square of a matrix expression

Let $A,C\in\mathbb{R}^{m\times n}$, $n\ge m$, $B\in\mathbb{R}^{n\times m}$, and $P$ be a real positive definite $m\times m$ matrix. Denote by $\mathcal{S}^n$ the space of $n\times n$ real symmetric ...
2 votes
1 answer
456 views

Constrained optimization over a trace functional

Let $A\in\mathbb{R}^{n\times n}$ be a stable matrix (i.e., the eigenvalues of $A$ have negative real parts). Consider the following optimization problem in $X \in \mathbb{R}^{n \times n}$ $$\begin{...
2 votes
1 answer
184 views

Condition for non-vanishing trace

Let $A$ and $B$ be two full column rank real matrices of dimension $n \times m$, where $n \ge m$. Let $P$ be an $m\times m$ positive definite matrix. Question: Does there always exist a symmetric $n \...
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? ...
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 ...
3 votes
1 answer
3k views

Solving a vector of quadratic equations

I have a system of $n \times 1$ equations $$ 0 = A\,vec(xx^t) + B x + C $$ where $x$ is a $n \times 1$ vector of unknowns $x^t$ means transpose $vec$ means $xx^t$ has been vectorized so has dimension ...
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 ...
1 vote
0 answers
305 views

how to find all the solutions to $I+A+\cdots+A^n=0.$ [closed]

Let $GL_3(\mathbb{Z}[i])$ be the group of invertible $3\times 3$ matrices whose coefficients are Gaussian integers.I want to find all the pair $(A\in GL_3(\mathbb{Z}[i]),n\in\mathbb{Z})$ satisfying $$...
0 votes
1 answer
666 views

Kernel of a projection

Given $m<n$. Suppose that $H$ and $K$ be $m \times n$ and $n\times (n-m)$ matrices such that rank$(H)=m$, rank$(K)=n-m$, and $HK=0$. For fixed non singular symmetric matrix $A$ define \begin{...