# Questions tagged [matrix-analysis]

The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.

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### Intersection of parameterized cosets in a free-abelian group

Let $L_1,L_2$ be subgroups of $\mathbb{Z}^m$, and let $\mathbf{P}_1,\mathbf{P}_2$ be $r\times m$ integer matrices. Then, it is straightforward to check that the set
\begin{equation}
S_{\{1,2\}} := \{...

**3**

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65 views

### How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$?

Suppose I have two matrices $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$. I want to know what's the best similarity transformation between these two matrices when we ...

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43 views

### How to show that a continuous family of symmetric matrices is uniformly positive?

My problem : I have a family of $4 \times 4$ symmetric matrices. More precisely consider an interger $d$, a real $\lambda> 0$ and define the family $S_{\lambda}$:
$ \{A(\lambda,x_1,x_2) ; (x_1,...

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25 views

### equivalent definition of k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...

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**1**answer

86 views

### Decomposition of one Matrix into six matrices [closed]

He folks, here's my problem:
Let $\mathbf{A}, \mathbf{B}, \mathbf{X}_1, \mathbf{X}_2, \mathbf{X}_3, \mathbf{Y}_1, \mathbf{Y}_2, \mathbf{Y}_3\in\mathbb{R}^{3\times 3}$ with determinante det()=+1. The ...

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**1**answer

110 views

### A generalization of invariant and coinvariant subspaces

Let $\mathcal{A}$ be a subalgebra of $M_n(\mathbb{C})$. Is there a characterization of (or at least a name for) orthogonal projections $P \in M_n(\mathbb{C})$ with the property that $PABP = PAPBP$ for ...

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106 views

### A limiting sequence of positive definite matrices

Let $A\in\mathbb{R}^{n\times n}$ be a matrix with eigenvalues having (strictly) negative real part. Let $X\in\mathbb{R}^{n\times n}$, $X\succ 0$, be a positive definite matrix and let $P\succ 0$ be ...

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219 views

### An inequality for matrix norms

Working on a problem in combinatorics I come up with the following inequality on matrix norms, which I checked it also numerically:
Let $A=(a_{ij})$ be a real symmetric $n\times n$ matrix with ...

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106 views

### Positive and trace-preserving transformations with a common fixed point of full rank

The following problem which has been on my mind for a while now arises from the realm of quantum information involving quantum channels with a common fixed point of full rank, as well as majorization ...

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43 views

### On the existence of fixed points of a matrix iteration

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e. all eigenvalues of $A$ have negative real part). Let $\succeq$ denote the standard partial order in the cone of positive semidefinite ...

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68 views

### Perturbation analysis for functional calculus with a gap condition

Consider a real interval $I$ and a symmetric matrix $M$ whose eigenvalues are at least at distance $\delta$ from the endpoints of $I$. For a symmetric matrix $A$, define $P_I(A)$ to be the matrix ...

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**1**answer

112 views

### Norm/trace of product inequality involving skew symmetric matrices

I wonder if the following inequality involving skew symmetric matrices is true:
Suppose that $B,C \in \mathbb{R}^{d \times d}$ are skew-symmetric matrices, and $\Sigma \in \mathbb{R}^{d \times d}$ ...

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261 views

### Maximum of a quantity for two normal orthogonal vectors in $\mathbb{R}^n$

Let's define for every pair of vectors $u,v\in\mathbb{R}^n$, a quantity as follows:
$$f(u,v) = \sum_{1\leq i,j\leq n}|u_iu_j-v_iv_j|.$$
I want to find:
$$M(n)= \max \{f(u,v): u,v\in \mathbb{R}^n, |u|=...

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**1**answer

48 views

### Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE.
The set of Schur stable matrices is
\begin{align*}
\mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\},
\end{align*}
where $\rho(\cdot)$ denotes ...

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**1**answer

54 views

### Efficient way to compute eigenvalue decomposition for following problem

I have an optimization problem
$$\begin{array}{ll} \text{minimize} & Tr(X^TAX) \\ \text{subject to} & X^TX=I
\end{array}$$
where $A\in R^{n \times n}$ and it is symmetric positive definite, ...

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**1**answer

84 views

### Matrix equation of the form $C A C^\intercal = D$

Consider the following square matrix
\begin{align}
A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\
0 & d & -d+1 & -\frac12 & 0 & ...

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**1**answer

173 views

### Can we classify the general linear maps such that for a fixed matrix $A \in M_n(\mathbb R)$ the conjuagation action fixes the first column?

Let $A \in GL_n(\mathbb R)$ be fixed. Let us consider the conjugation action by $G \in GL_n(\mathbb R)$, i.e., $G^{-1}AG$. I would like to see a way to identify the matrices such that the action fixes ...

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90 views

### Geometric mean of three or more positive definite matrices

The geometric mean of two positive definite (Hermitian) matrices of same size is defined by $$A\natural B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},$$equivalently,
$$A\natural B =(BA^{-1})^{1/2}A=A(A^...

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12 views

### Every singular DNN realization of $G$ is completely positive implies

DNN denotes doubly non-negative matrices(both entry wise non-negative and is positive semi-definite). Let $G$ be a Graph.
The following two are equivalent:
(a) Every non-singular DNN realization of $...

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87 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 $A$...

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253 views

### A (linear) optimization problem subject to (linear) matrix inequality constraints

Let $A \in \mathbb{R}^{n \times n}$ be a Hurwitz matrix, i.e. $A$ satisfies $\mathrm{Re}\,\lambda_i< 0$, where $\{\lambda_i\}_{i=1}^n$ is the set of eigenvalues of $A$. Suppose that the trace of $A$...

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119 views

### Calculate the rank of a symmetric matrix

I am interested in calculate the rank of symmetric matrices. Currently I use rankMatrix of the Matrix package of R to calculate it. However it is slow. I was thinking that it might be due to not using ...

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**2**answers

125 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 ...

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**1**answer

47 views

### The width of the minimum gap of an interpolated matrix

Question originally posted here, but I'm more likely to get an answer on MO. I'm looking for a reference on the following topic, should one exist. (Or a firm "stop looking and do it yourself".)
Let $...

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658 views

### Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the ...

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84 views

### An inequality concerning the solution of a Lyapunov equation

Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...

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**1**answer

48 views

### Bounding/approximating the largest eigenvalue of the special case of companion matrix

Suppose I have the following companion matrix ($d\times d$)
The companion matrix A. $1 \geq p \geq q \geq 0$. Let $x$ ($d\times 1$) be the all one vector and my underlying problem is to analyze the ...

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156 views

### About product of PSD matrices

In Theorem 3 in this paper, https://core.ac.uk/download/pdf/82822897.pdf, ``On a product of positive semidefinite matrices, A.R. Meenakshi, C. Rajian, Linear Algebra and its Applications, Volume 295, ...

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81 views

### On the existence condition of a solution to a matrix equation

When we analysis the observability of a dynamic system, we meet a question as follows:
Let $\bar{C}=[(CA^{-i})^{T} \; (CA^{-i+1})^{T} \; (CA^{-i+2})^{T} \cdots (CA^{-i})^{T}]^{T}$ and
\begin{...

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**1**answer

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 ...

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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 ...

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137 views

### When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE.
Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...

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**1**answer

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)-(...

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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\...

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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 ...

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**3**answers

808 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(...

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286 views

### Is the set $\{\zeta: \rho(A(\zeta))< 1\}$ connected for matrices under parameterization of first $m$ rows?

$\newcommand{\eqqcolon}{=\mathrel{\vcenter{:}}}$
Fix $n, m \in \mathbb N$ with $n > m$. Let $\zeta \in \mathcal{M}(m \times n; \mathbb C)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \...

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88 views

### Is the closure $\overline{ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) < 1\} }$ equal to $ \{X \in \mathbb{R}^{m \times n} : \rho(M-NX) \le 1\}$

I am not sure whether this question fits this forum (I will delete it if not appropriate here). But I asked this on MSE over a week ago with no answer and then put a bounty still got no answer. Here ...

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**1**answer

196 views

### Square root of a large sparse symmetric positive definite matrix

I am trying to calculate
$$Y = A^{\frac 12} X$$
where $A$ is a very large and sparse positive definite matrix, say, $10^4 \times 10^4$. Matrix $X$ is known and, say, $10^4 \times 100$. Is there any ...

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**1**answer

147 views

### when is an eigenvalue differentiable with respect to a parameter? [duplicate]

Let say we have a symmetric matrix $A(\omega)$ depending smoothly on some variables $\omega \in \Omega$ with $\Omega \subset \mathbb{R}^d$ a $d$-dimensional parameterspace (this means the eigenvalues ...

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169 views

### Is this parametrized semidefinite program convex?

I am considering an optimization problem of the form:
\begin{equation}
\begin{split}
f(s) &= \min_{X} \mathrm{tr}(C(s)X) \\
&\;\;\;\;\;\;\;\;\;\;\; X \ge 0, \\
&\;\;\;\;\;\;\;\;\;\;\; \...

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**1**answer

339 views

### Well known matrix inequality?

I suspect that the following matrix inequality is well known, but I can't find a reference or proof:
Given $n \times n$ symmetric matrices $A,B$ such that $I_n \leq A,B$, is the following true?
$${...

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173 views

### Diagonal elements of Hermitian matrices with given eigenvalues

Given real vectors $d = (d_1, \ldots, d_n)$ and $\lambda = (\lambda_1, \ldots, \lambda_n)$, where I will assume that their coefficients are arranged in non-increasing order, the Schur-Horn theorem ...

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25 views

### Depth bound of generating an matrix algebra

Let $X$ be Hilbert spaces $\mathbb{C}^d$, and $L(X)$ be the sets of linear operators of $X$. We are given a matrix subspace $S\subset L(X)$. Via the following procedure, one can generate the smallest ...

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111 views

### L1 norm constraint on product of 2 matrix

I want to solve below minimization problem
\begin{equation*}
\begin{aligned}
& \underset{A, B}{\text{minimize}}
& & ||Y-AB^T -D||_F^2 \\
& \text{subject to}
&& |A_i|_1 \leq a,...

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106 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 ...

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74 views

### Decay rate of least eigenvalue of Gram matrices

Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$:
$$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$
In ...

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80 views

### The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$

The algebraic connectivity of $P_n$, the path on $n$ vertices does not exceed $\frac{12}{n^2-1}$.
Let algebraic connectivity of $P_n$ be denoted by $\mu$. I have proved a result that if $G$ is a ...

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**1**answer

74 views

### On an error bound for matrix constraints

Given $A,B\in\mathbb R^{n\times n}$ such that there is an $U\in\mathbb R^{n\times n}$ with $UU'=I$ and $UAU'=B$.
Suppose each entry of $V$ is within $\pm\epsilon$ of each entry of $U$ is there a way ...

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**1**answer

84 views

### SVD of two matrices A and B having the same right singular vectors?

I saw this statement in a lecture note
Assume the generalized SVD of matrices $A\in R^{m\times n}$ and $B\in R^{p\times n}$ given as:
$$U^TAX = diag(\alpha_1, ..., \alpha_n),~ U^TU = I_m$$
$$...