All Questions
Tagged with linear-algebra matrix-analysis
364 questions
1
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1
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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 ...
- 39.9k
6
votes
1
answer
886
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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 ...
- 52.3k
5
votes
1
answer
319
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:
$$ \Vert A \Vert_* = \sqrt{\operatorname{tr}(...
- 108k
1
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0
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163
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An estimation of the largest eigenvalue of a submatrix of $\left(\cos(\frac{kl\pi}{4n})\right)_{k,l=1}^n$
Let us consider the following matrix $A=(a_{k,l})$ where
$$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\...
- 4,058
2
votes
0
answers
88
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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
3
answers
3k
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Spectral properties of the LDL^T matrix factorization
Assume that a square, symmetric matrix $A$ can be factored into $A=LDL^T$ where $L$ is unit lower triangular and $D$ is diagonal. For indefinite $A$, $D$ may have $2x2$ blocks on the diagonal. How ...
- 4,943
2
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0
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43
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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 ...
- 4,058
2
votes
0
answers
46
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Combining SVD subspaces for low dimensional representations
Suppose we have matrix $A$ of size $N_t \times N_m$, containing $N_m$ measurements corrupted by some (e.g. Gaussian) noise. An SVD of this data $A = U_AS_A{V_A}^T$ can reveal the singular vectors $U_A$...
- 21
4
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0
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990
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Lower bound minimum eigenvalue of a positive semi-definite Hermitian matrix with bounded entries
Let $M \in \mathbb{C}^{n \times n}$ be a matrix with the following properties:
$M$ is Hermitian and positive semi-definite (all the eigenvalues are real and nonnegative).
The diagonal entries of $M$ ...
- 41
2
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0
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107
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Gradient of QZ decomposition
Let $A$ and $B$ be an $m \times n$ matrix of rank $ k_1 \le \min(m,n) $ and $ k_2 \le \min(m,n) $. Then the QZ decomposition or the generalized Schur decomposition is $A = USV^T$ and $B = UTV^T $, ...
- 61
0
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0
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174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 63.9k
8
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0
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232
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Decay of orthogonal contributions in a random set of vectors
Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
1
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0
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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 ...
0
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0
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114
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Degeneracies in linear combination of tensor product of Pauli matrices
Let $P_i \in \{I,X,Y,Z\}^{\otimes n} $, that is $P_i = \bigotimes_{i =1 }^n \sigma_i$ with $\sigma_i \in \{I,X,Y,Z\}$, where
$$
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \hspace{1cm} X =...
- 63.9k
2
votes
1
answer
178
views
Define a matrix square root that preserves regularity
Let $A:\mathbb{R}\to \mathbb{R}^{n\times m}$ and $B\in \mathbb{R}^{n\times k}$. Is it possible to define $C:\mathbb{R}\to \mathbb{R}^{n\times m}$ satisfying the following two properties:
for all $t\...
- 503
1
vote
0
answers
53
views
Matrix powers up to multiplicative factor
Let $A$ be a real $n\times n$ matrix, $A_n = A^n$, and
$$ \bar A_n = \lbrace\alpha A_n, \alpha\in \mathbb{R}\rbrace.$$
I am interested in characterizing the behavior of $\bar A_n$ when $n\rightarrow \...
- 401
0
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1
answer
365
views
How to estimate the norm of a matrix
There is a matrix as following,
\begin{eqnarray}
A = \left (
\begin{array}{l}
0 \quad \quad \quad \quad \quad \quad \quad ~~ 1\\
b \quad ~~~0 \quad \quad \quad \quad \quad a\\
ab \quad ~~ b \...
CommunityBot
- 1
6
votes
1
answer
290
views
Recover approximate monotonicity of induced norms
Let $A$ some square matrix with real entries.
Take any norm $\|\cdot\|$ consistent with a vector norm.
Gelfand's formula tells us that $\rho(A) = \lim_{n \rightarrow \infty} \|A^n\|^{1/n}$.
Moreover, ...
CommunityBot
- 1
0
votes
0
answers
180
views
Adding the AWGN to the data makes its covariance matrix always positive definite?
I'm working on a numerical method that estimates direction-of-arrivals in antenna arrays.
I realized that every time I add the AWGN (Additive white Gaussian noise) to a data (which is a matrix), its (...
2
votes
1
answer
141
views
On the eigen vectors of a diagonalizable matrix
Let us consider the space $M_n(\mathbb{C})$. By a unitary matrix $U=(u_{ij})$ we mean that $U^{-1}=(\overline{u_{ji}})$.
Q. Let $U$ be a unitary matrix. I am looking for the pairs of matrices $(D,A)$ ...
2
votes
0
answers
300
views
Eigenvector of Hadamard matrix functions
Let $X\in\mathbb{R}^{n\times n}$ with SVD $X=UDV^T$. Are there known results regarding the eigenvectors of $Y=X^{\odot g}$? I am mainly interested in simple functions such as $g(z)=z^2$, i.e. $Y_{ij}=...
- 105
2
votes
2
answers
104
views
Inequality for matrix with rows summing to 1
Let $A$ be real matrix with $M > 1$ rows and $K > 2$ columns, and each entry $a_{m,k} \in (0,1)$, with each row summing to $1$. For all $m$
$$
\sum_{k=1}^{K} a_{m,k} = 1
$$
I want to find out if ...
- 20.2k
0
votes
0
answers
105
views
Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces
Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
- 31
4
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1
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720
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Singular value decomposition of truncated discrete Fourier transform matrix
Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that
\begin{align}
F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N.
\end{align}
What we can say about the singular value ...
- 188k
3
votes
0
answers
275
views
Schur-Horn theorem for principal submatrices
The Schur-Horn theorem says that there exists a Hermitian matrix with diagonal entries $d_1,d_2,\ldots,d_n$ and eigenvalues $\lambda_1,\lambda_2,\ldots,\lambda_n$ if and only if $(\lambda_1,\lambda_2,\...
- 6,351
1
vote
1
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167
views
Second order matrix differential equation in the space of symmetric positive definite matrices
In the construction of interpolations in the space of Gaussian measures, I encountered a second order matrix differential equation in the set of symmetric positive definite matrices $\mathbb{S}_+^d\...
- 128k
2
votes
1
answer
74
views
Limitation through the singular values
Given matrix $X \in \mathbb{R}^{m\times n}$ and sequence $\left\{X^k\right\}_k$ converges to $X$ according to the Frobenius norm. I wonder that $\sigma_i(X^k)$ converge $\sigma_i(X)$ or not (where $\...
- 20.2k
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 33.8k
0
votes
0
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198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 188k
4
votes
0
answers
163
views
Matrix logarithm of unitary factor from polar decomposition of product of positive definite matrices
This question is crossposted from Math Stackexchange here. I crosspost without much edits as I think this is the best way to phrase the question and because I received no feedback on the original post ...
- 41
2
votes
1
answer
448
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How to find upper and lower bound
Let $\Sigma \in S_{++}^n$ be a symmetric positive definite matrix with all diagonal entries equal to one. Let $U \in \mathbb{R}^{n \times k_1}$, $W \in \mathbb{R}^{n \times k_2}$, $\Lambda \in \mathbb{...
CommunityBot
- 1
7
votes
1
answer
511
views
Is it true that $\lVert A\rVert \leq \lVert A^2\rVert$ for $A\in \operatorname{SL}(2, \mathbb{R})$ when $\operatorname{trace}(A)>2$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\trace{trace}$Let $A \in \SL(2,\mathbb{R})$ and $\trace(A)>2$. Is it true that $$\lVert A\rVert \leq \lVert A^2\rVert,$$
where $\lVert \rVert$ is the ...
- 24.2k
11
votes
4
answers
5k
views
The derivative of the Cholesky factor
Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
- 63.9k
1
vote
1
answer
106
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Matrix equation with projection matrix
I need to solve the following equation for $P \in\mathbb{R}^{r\times d}$
$$P - G_1G_2(\lambda P^\top(PAP^\top)^{-1}P + A^{-1} ) = 0,$$
where the other quantities are known: $A\in\mathbb{R}^{d\times d}$...
- 188k
49
votes
3
answers
4k
views
Is this proof of Perron's theorem correct, and if so is it original?
A few years ago, I came up with this proof of Perron's theorem for a class presentation:
https://pi.math.cornell.edu/~web6720/Perron-Frobenius_Hannah%20Cairns.pdf
I've written an outline of it below ...
- 5,146
5
votes
0
answers
836
views
Gershgorin's 2nd theorem (disjoint circles): elementary proof?
Let $A \in \mathbb{C}^{n\times n}$ be a complex matrix. We let $a_{i,j}$ be the $\left(i,j\right)$-th entry of $A$ for all $i, j \in \left[n\right]$ (where $\left[n\right]$ denotes $\left\{1,2,\ldots,...
- 33.8k
0
votes
0
answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k
5
votes
1
answer
2k
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Diagonalization of real symmetric matrices with symplectic matrices
Consider the following real symmetric matrix
$M=\left[\begin{array}{ccc} A & B\\ B^T & D \end{array}\right]$
Both A and D are real symmetric $n\times n$ matrices. B is a real $n\times n$...
0
votes
1
answer
274
views
Does positivity of the n(n-1)/2 principal minors formed from 2 x 2 submatrices ensure positive-definiteness of the n x n matrix itself?
I am interested in conditions under which an $n \times n$ matrix ($\rho$) is positive definite. Of course, one necessary and sufficient set of conditions is that the $n$ leading minors of $\rho$ each ...
- 25.7k
0
votes
0
answers
46
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Lipschitz solutions to linear complementarity problems (LCP)
Let $M\in\mathbb{R}^{n\times n}$.
For $q\in\mathbb{R}^n$, define the set:
$$S_M(q)=\{y\in\mathbb{R}^n|y\ge 0,q+My\ge 0, y^\top (q+My)=0\}.$$
This is the set of solutions to the LCP $(q,M)$.
We say $...
- 183
0
votes
0
answers
194
views
Rewriting Kronecker product
im considering a pole placement problem in control theory and my controler has a specific form:
$$R=I_n\otimes q$$
where $I_n$ is the identitiy matrix of size $n$ and $q\in\Re^k$ is a vector of the ...
- 188k
1
vote
2
answers
143
views
If $x \ge 0$ and $\mathbf{1}^Tx \le \|x\|^2$ then $\mathbf{1}^T(I - xx^T / \|x\|^2) \mathbf{1} \ge \| [\mathbf{1} - x]_+ \|^2$
Notation. Denote $\mathbf{1}=(1,1,\ldots,1)$ as the vector-of-ones in $\mathbb{R}^n$. Write the "positive part" as $[\alpha]_+ = \max\{\alpha,0\}$ for $\alpha\in\mathbb{R}$ and $[(x_1,x_2,\...
- 570
3
votes
1
answer
741
views
Operator norm of difference of matrix decompositions
This question is in part related to a question that I have already posed.
Say I have two symmetric positive definite matrices and their respective Cholesky decompositions $\mathbf{A} = \mathbf{L}_A \...
- 52.3k
1
vote
0
answers
70
views
Projecting matrix from LDL^T factorization
When factorizing a real symmetric matrix $A$ into $LDL^T$, the matrix $D$ can have 1x1 or 2x2 blocks on the diagonal. A condition for $A$ to be positive-definite is that all 1x1 and 2x2 blocks of $D$ ...
- 111
2
votes
0
answers
176
views
System of matrix equations
Problem definition: Let $x_i \in \mathbb{R}^d$ and $a_i \in [0,1]$, for all $i = 1,\dots, k$ (with $k\geq d$). Define $M(a) = \sum_{i = 1}^k a_i x_ix_i^T,$ and assume $M(a) \succ 0.$
Question: Is ...
- 245
-2
votes
1
answer
1k
views
Derivative of log determinant [closed]
Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right).
$$
- 245
1
vote
1
answer
64
views
Upper bound on the size of vectors contained in an ellipsoid?
Crossposted at Math SE
Consider the diagonal matrix
$$D=\left[\begin{array}{cccc}
1^{- 2 p} & 0 & \cdots & 0 \\
0 & 2^{-2 p} & \cdots & 0 \\
\vdots &...
- 109k
1
vote
1
answer
201
views
Matrix equation involving quadratic form
Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let
$$A_i(\alpha) = (X\Lambda(\alpha) X^...
- 188k
3
votes
0
answers
2k
views
Multiplication of two Pauli string
Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...
- 131
6
votes
2
answers
259
views
Distance of low-rank matrices to the identity for the $\infty$-norm
I am trying to get a lower bound (or even the exact value) of
$$
\min_{X \in \mathbb{R}^{n\times n}} \|X - I_n\|_{\infty} \enspace \text{s.t.} \enspace \mbox{Rank}(X) = m
$$
where $m \leq n$, and the ...
- 9,486