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Results tagged with matrices
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user 62673
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.
3
votes
1
answer
368
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Closed-form expression for differential of matrix function
Let $X$ be a real $n\times n$ positive semidefinite matrix of rank $m\le n$ and let $Y\in\mathbb{R}^{m\times n}$ be the unique matrix satisfying (i) $X=Y^\top Y$, and (ii) $Y\, [I\, |\, 0]^\top = L$ w …
- 2,712
4
votes
1
answer
376
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On the solvability of a matrix equation
Let $\{C_i\}_{i=1}^N$ be a set of $n\times m$ real matrices of full column-rank and such that $\mathrm{Range}[C_1,C_2,\dots,C_N]=\mathbb{R}^n$, $\{P_i\}_{i=1}^N$ a set of $m\times m$ positive definite … real matrices. …
- 2,712
4
votes
1
answer
223
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$\mathrm{diag}\left[(A+D)^{-1}\right] \ge \left[\mathrm{diag}(A)+D\right]^{-1}$?
Let $A\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix and $D\in\mathbb{R}^{n\times n}$ be a diagonal positive definite matrix. Let $\mathrm{diag}(X)\in\mathbb{R}^{n\times n}$ denote the d …
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0
votes
1
answer
93
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On the (qualitative) behavior of a coupled differential equation
Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following differential e …
- 2,712
1
vote
0
answers
37
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Attractivity of a system with state-dependent transitions
Let $A\in\mathbb{R}^{n\times n}$ and consider the following dynamical system:
$$
\frac{\mathrm{d}x(t)}{\mathrm{d}t} = -x(t)+\max\{0,Ax(t)\}, \ \ \ \ x(0)\in\mathbb{R}^n,
$$
where $\max\{\cdot\}$ acts …
- 2,712
13
votes
2
answers
1k
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A log inequality for positive definite trace-one matrices
Let $\{v_i\}_{i=1}^N$ be a set of $n$-dimensional real vectors and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. I would like to prove (or disprove) the following in …
- 2,712
2
votes
0
answers
50
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Metrics on the group of unimodular polynomial matrices
The group of unimodular matrices $\mathbb{U}[s]^{n\times n}$ is given by the set of $n\times n$ square (real) matrix-valued polynomials $\mathbb{R}[s]^{n\times n}$ which admit a polynomial inverse. … the space $\mathbb{U}[s]^{n\times n}$ which, in some sense, resembles to the "natural", i.e. affine-invariant, metric of the general linear group $\mathbb{GL}(n)$ for the case of $n\times n$ constant matrices …
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8
votes
1
answer
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A generalized log inequality for positive definite trace-one matrices
Let $\{V_i\}_{i=1}^N$ be a set of $n\times m$, $n\geq m$, real matrices of full column rank and let $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive definite trace-one matrix. …
- 2,712
9
votes
1
answer
840
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Maximizing a ratio of determinants
Consider the optimization problem over the set of positive semidefinite matrices with trace less or equal than one
$$
\max_{A\in\mathbb{R}^{n\times n}\,:\,A\geq 0,\, \mathrm{tr}(A)\leq 1}\frac{\det(A+\ …
- 2,712
3
votes
1
answer
267
views
Solving a "reversed" Stein equation
Let $P$ and $Q$ be positive definite matrices. Consider the following matrix equation
$$\label{star}\tag{$\star$}
XPX^\top - P = -Q, \quad X\in\mathbb{R}^{n\times n}.
$$
My question. …
- 2,712
1
vote
0
answers
105
views
Primitivity of $AA^\top$
Here is my question: Have graphs whose adjacency matrices satisfy condition C been studied in the literature? If so, do such graphs have a name?
Any answer/comment would be highly appreciated. …
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4
votes
0
answers
278
views
Maximizing a certain eigenvalue ratio
Let $A\in\mathbb{R}^{n\times n}$ be an Hurwitz stable matrix (i.e., the spectrum of $A$ lies on the left-half complex plane) and let $P$ be the unique positive definite solution of the following Lyapu …
- 2,712
5
votes
1
answer
152
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Finding a particular matrix factor
Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a factorizat …
- 2,712
1
vote
2
answers
1k
views
A "positive diagonal plus skew-symmetric" matrix decomposition
Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly positive eigenvalues (note that $A$ is not required to be symmetric).
My question. Do there exist an orthogonal …
- 2,712
4
votes
0
answers
248
views
Is every stable matrix orthogonally similar to a $D$-skew-symmetric matrix?
Let $\mathrm{diag}(A)$ denote the diagonal matrix with diagonal entries of $A\in\mathbb{R}^{n\times n}$ and let $\succeq$ denote the standard partial order in the cone of (symmetric) positive definite matrices … "On two classes of matrices with positive diagonal solutions to the Lyapunov equation." Linear algebra and its applications, 59 (1984): 19-27. …
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