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.
697
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49
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When is the sum of matrices (circulant + [super upper triangular]) not diagonalizable?
By the circulant matrix $C \in M_n(\mathbb{R})$, we mean that
$$ C = \left[\begin{array}{c|c|c|c} e_n & e_1 & \cdots & e_{n-1} \end{array}\right] $$
where $e_1,\dots,e_n$ are the standard ...
- 3,774
4
votes
1
answer
101
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Sum of holomorphic squares?
Consider a variable $z \in \mathbb{R}^n$ and assume $u(z) \in \mathbb{R}^m$ and $H(z) \in \mathbb{R}^{m \times m}$. Further assume that $H(z)$ is symmetric positive definite for every $z$. Consider ...
- 695
2
votes
1
answer
96
views
The eigenvectors of adding a particular rank one matrix to the circulant matrix
Suppose that $e_1, \cdots, e_n$ are the standard vectors of the Euclidean space $\mathbb{R}^n$.
Let us consider the backward shift operator $T:\mathbb{R}^n\to \mathbb{R}^n$ given by $Te_k=e_{k-1}$ ...
- 3,774
5
votes
1
answer
230
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Diagonalization of symmetric matrices of functions
I asked this question some time ago in MSE but I didn't recieved any feedback.
https://math.stackexchange.com/questions/4672664/diagonalization-of-symmetric-matrices-of-functions
This problem arised ...
- 143
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118
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Maximizing the norm of a sum of Hermitian matrices
Consider the following problem:
Problem: Given $n\times n$-Hermitian matrices $A_1,\dots,A_r$, find $e_1,\dots,e_r\in\{-1,1\}$ such that $\|e_1A_1+\dots+e_rA_r\|_\infty$ is maximized. Here the norm is ...
- 25.5k
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2
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164
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On a matrix trace inequality
Let $\mathcal H(n)$ be the set of $n\times n$ Hermitian matrices, and $\mathcal S(n) \subset \mathcal H(n)$ be the subset of density matrices, i.e., $A \in \mathcal S(n)$ iff $A$ is Hermitian, ...
- 469
3
votes
1
answer
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Condition for 3×3 block matrix to be stable
Given a square symmetric matrix $H\in\mathbb{R}^{n\times n}$, design a symmetric positive definite matrix $M\in\mathbb{R}^{n\times n}$ and positive scalar $\alpha$ such that the following ${3n\times ...
- 33
1
vote
1
answer
76
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Norm inequality
In an article I read, I have the following inequality:
$\|A-B\|_1 \geq \max \{ \|A 1_m- B 1_m \|_1, \|A^T 1_n - B^T1_n\|_1 \}$
Where $A, B \in \mathbb{R}_+^{m\times n}$.
The $\|\cdot\|_1$ refers ...
- 13
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votes
0
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84
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Modeling decay of a linear system with a mixing term
I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence
$$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}...
- 2,118
3
votes
1
answer
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Positive definiteness of a matrix-valued function
This question is a repost from math.se, where I didn't receive an answer.
Are there simple conditions on an $d \times d$ matrix B under which
$$
f(t, s)
=
\begin{cases}
\exp(-B |t - s|^\alpha), &...
- 109
2
votes
1
answer
140
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Question on density of certain set of matrices
Let $B$ be an invertible real matrix and let $Q=\{A \text{ real}\mid AB^{T} \text{ is symmetric}\}$. Is the subset $S=\{ A \in Q\mid A+A(B^{-1}A)^{2} \text{ is symmetric}\}$ of measure zero in $Q$? I ...
- 151
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votes
1
answer
384
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Approximating sum of entries of $\exp(A-B)$ for diagonal $A$ and rank-$1$ $B$?
I have non-negative $d\times d$ matrices $A$, $B$ and need a tractable way to compute the sum of all entries of $\exp(-t(A-B))$ where $A$ is diagonal and $B$ symmetric rank-$1$. IE
$$f(t)=\langle\exp(-...
- 2,118
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0
answers
41
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Finding gradient w.r.t. matrix inputs
Suppose that $A \in \mathbb{R}^{1\times \ell}$, $B \in \mathbb{R}^{\ell \times m}$ are fixed matrices and ${\bf z} \in \mathbb{R}^d$ is a fixed vector. Let $V \in \mathbb{R}^{m\times k}$, $B \in \...
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The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and
$$a_{1,2}=\cdots=a_{n-1,n}=1$$
Any approach to ...
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votes
0
answers
60
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Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
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40
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Subspace contains two matrices with the same spectrums
Consider matrix subspace $\mathbb{C}^{m\times n}$.
What is the minimum $d$ such that for any $d$-dimensional matrix subspace of $\mathbb{C}^{m\times n}$, there exists two distinct matrix $M,N$ such ...
- 1,453
-1
votes
1
answer
202
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How to compute the spectral norm of this matrix
Consider $$\left\|2\sum_{i<j}L_{ij}+4\sum_i \operatorname{diag}e_i \right\|,$$ where
(1) $L_{ij}=\operatorname{diag}e_i+\operatorname{diag}e_j-e_ie_j^T-e_je_i^T$
(2) $e_i$ denotes $n$-by-$1$ vector ...
- 53
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0
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62
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Follow-up question regarding real singular matrices with additional details
After my question whose answer turned out to be false, I re-examined the course of my proof, which is actually seperate from the one in my question, and found out that there's another condition, at ...
- 151
1
vote
2
answers
133
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Singularity of matrix pencil-like expression
I was working more on the topic on my previous question when I have to know whether the following statement is true to circumvent the "exception" caused by division by singular matrices; ...
- 151
3
votes
1
answer
503
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Is the set of real matrices with at least one real logarithm closed under multiplication?
Let $S$ be the set of real matrices with at least one real logarithm. For some couple of its elements, for example those with at least (one real logarithm each with submultiplicative norm smaller than ...
- 151
1
vote
0
answers
100
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Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 151
1
vote
0
answers
196
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Inequality on matrix trace
Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems :
$$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma ...
- 469
1
vote
0
answers
139
views
Equivalent definition of positive semidefinite
I am reading a paper which repetitively uses the following statement, but I don't know why this is true:
Statement Let $A$ be a symmetric $n$-by-$n$ matrix, and $B$ be a $n$-by-$n$ matrix, $A\geq 0$ ...
- 53
1
vote
1
answer
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Control the summation of a diagonal matrix and another matrix to be full rank
Statement. To ensure the rank of $\operatorname{ddiag}(AQQ^T)-\sigma\Delta=n$, it is sufficient to require $\min_i(\operatorname{diag}(AQQ^T))_i>\sigma\lVert\Delta\rVert$.
Note: $Q\in\mathbb{R}_{n\...
- 53
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votes
1
answer
69
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$
Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix.
$$...
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3
votes
0
answers
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Recovering the matrix when the Schur decomposition of its blocks are known
Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and
$$E=\left(\begin{array}{cc}
G & X \\
X^t & H
\end{array}\right)$$
where $G,H,X$ are $m\times m$ matrices.
Suppose that $...
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votes
1
answer
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Is the set of purely real square matrices, that are complex-diagonalisable, dense in the set of real matrices?
A quick search for "diagonalisable matrix" on Wikipedia immediately gives the result that the set of real-diagonalisable matrices is not dense in the set of real matrices.
I need, however, ...
- 151
1
vote
1
answer
75
views
Eigenvalues of a circulant: DFT or Inverse DFT Convention?
Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
- 741
2
votes
1
answer
83
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Generate a low-rank sparse covariance matrix
May I ask how to generate a low-rank sparse covariance matrix? Thank you!
- 21
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votes
0
answers
67
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Reference for (general case) of uniqueness of singular value decomposition (SVD)
My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values.
I believe that the statements and proofs on this StackExchange posts are ...
1
vote
1
answer
109
views
Third order matrix differential norm
Suppose we have a function $f:\mathbb{R}^n\to\mathbb{R}$ that is at least three times differentiable. Clearly, there is a relationship between the symmetric trilinear form $$T_1=\nabla^3f(x),$$ and ...
- 39
3
votes
1
answer
115
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On the bounds of the sum of the squares of spectral variation of two real symmetric matrices
Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
- 61
3
votes
2
answers
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Extend an inequality on matrix norms
Let $A$ denote an $n \times n$ matrix, and $\sigma_i(\cdot)$ denote $i$-th largest singular value. Can we extend the following result to general $p \geq 1$?
For all $k = 1, \dots, n$,
$$ \sum_{i = 1}^...
- 153
3
votes
0
answers
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Can the Jordan decomposition of a matrix be computed in a backwards stable way?
Let $PJP^{-1}$ denote the Jordan decomposition of $M$. The matrix $J$ is a direct sum of Jordan blocks; it is unique up to permutation of the Jordan blocks. The matrix $P$ is not unique.
There are two ...
- 4,511
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votes
1
answer
99
views
Matrix inequality in a paper by Piccinini-Spagnolo
In the paper 'On the Holder continuity of solutions of second order elliptic equations in two variables' by Piccinini and Spagnolo, they prove the following estimate:
$$
\begin{array}{ll}
\left(\int_S ...
- 473
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votes
2
answers
402
views
What is the inverse of a triangular matrix whose nonzero elements are binomial coefficients? What is the closed-form solution to a recursive relation?
Let
\begin{equation*}
\begin{split}
M_m
&=\begin{pmatrix}
-\binom{1}{0} & \binom{2}{0} &-\binom{3}{0} &\dotsm & (-1)^{m-1}\binom{m-1}{0} & (-1)^m\binom{m}{0}\\
0 & \binom{2}...
- 706
1
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0
answers
49
views
Solve linear matrix equation involving convolution
I am facing following equation:
$$
A * X + C \cdot X = D
$$
with:
$A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure,
$X \in \mathbb{R}^{n \times n}$ the ...
- 11
1
vote
1
answer
108
views
Bounds on the Bures–Wasserstein distance
The Bures–Wasserstein distance between $n\times n$ positive semidefinite matrices $A$ and $H$ is defined to be
$$
d(A,H) := \left[ \operatorname{tr} A + \operatorname{tr} H - 2\operatorname{tr} (A^{1/...
- 233
1
vote
0
answers
63
views
Diagonal entries and determinant of positive definite matrices
Given a $3x3$ symmetric, positive matrix $\lambda |x|^2 \leq x^TAx \leq \Lambda |x|^2$, let us denote $a_{11}$, $a_{22}$ and $a_{33}$ to be the diagonal elements. Furthermore, let us denote the values ...
- 473
1
vote
1
answer
61
views
A question about the sign of quadratic forms on nonnegative vectors
Let $M$ be a real square matrix of order $n\ge 3$.
Assume that for every nonnegative vector $\textbf{z}\in \mathbb R^n$ which has at lease one zero entry we have $\textbf{z}^T M \textbf{z} \ge 0$.
Can ...
- 13
9
votes
1
answer
307
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One question on circulant $(-1,1)$-matrices
Let $n > 13$ be a positive integer. Is there any $n\times n$ circulant $(-1,1)$-matrix $A$ satisfying the following property:
$$AA^T=(n-1)I+J$$
where $I$ is the $n\times n$ identity matrix and $J$ ...
- 220
3
votes
1
answer
179
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One question on block-circulant matrices
Circulant matrices are very useful in digital image processing.
I found the general formula for determinant of circulant matrix.
But I think it is not suitable for block-circulant matrices.
For ...
- 220
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0
answers
96
views
The upper bound for the eigenvalues of a real symmetric matrix
Let $A$ be a symmetric matrix in $M_n(\mathbb{R})$. Except Gershgorin circle theorem, is there any other theorem to say us about the upper bound of the eigenvalues of $A$?
- 3,774
1
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0
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96
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An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$
Let us consider the following matrix $A=(a_{k,l})$ where
$$A=\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$$
Let us consider the submatrix $A_0$ of $A$ whose entries are those $a_{k,l}$ where $k\...
- 3,774
2
votes
0
answers
69
views
Nuclear norm minimization of convolution matrix (circular matrix) with fast Fourier transform
I am reading a paper Recovery of Future Data via Convolution Nuclear Norm Minimization. Here, I know there is a definition for convolution matrix.
Given any vector $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)^...
- 21
5
votes
1
answer
197
views
Analytical form for the nuclear norm of an $n \times n$ matrix
I get the follow equation in a paper. Let $A \in \mathbb{R}^{2 \times 2}$, then $M = A^TA$ is a positive semi-definite matrix, the nuclear norm of $A$ is:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
- 53
2
votes
1
answer
71
views
Distribution of scaled Johnson-Lindenstrauss transforms
Suppose that $\mathcal{D}$ is a Johnson-Lindenstrauss (JL) distribution on $\mathbb{R}^{r\times n}$ ($1 \le r \le n$), meaning that there exist constants $\epsilon, \delta \in(0,1)$ such that
$$
\...
- 221
0
votes
0
answers
105
views
$\mathbb{P}(\|A^Tx\| \ge \epsilon \|A\|\|x\|) \ge \delta$ for all $x \in\mathbb{R}^n$
Let $r$ and $n$ be integers such that $1 \le r \ll n$, and $\|\cdot\|$ denote the Euclidean norm of vectors or the spectral norm of matrices.
Suppose that $\mathcal{D}$ is a probability distribution ...
- 221
0
votes
0
answers
71
views
A good estimation for the norm of a matrix
For a given natural number $n$, let us consider $E_n=\{0\leq k\leq n-1 : k\equiv_41 \}$. Suppose that $E_n$ including $k_1<k_2\cdots < k_m$. Consider the following matrix $A$:
$$A=\left(\cos\...
- 3,774
2
votes
0
answers
35
views
Selecting some linearly independent columns of a particular matrix
Let us consider the matrix $C=A_1+A_2$ where :
$A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$
$A_2$ is the the $n$ by $n$ block ...
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