All Questions
Tagged with linear-algebra matrix-analysis
364 questions
2
votes
0
answers
107
views
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
votes
0
answers
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 \...
- 91
8
votes
0
answers
232
views
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 $v$:
$$\frac{v_1}{\|v_1\|},\...
- 2,252
1
vote
0
answers
146
views
Identities for the determinant of a matrix similar to $\det(A)=\exp\circ\operatorname{tr}\circ\log(A)$ for different matrix functionals
The identity for the determinant of $A$ in the title is well know in matrix analysis and comes from the Jacobi's formula. I am interested in the existence of nontrivial formulas like this one (they do ...
- 573
0
votes
0
answers
114
views
Degeneracies in linear combination of tensor product of Pauli matrices
Let $P_i \in \{I,X,Y,Z\}^{\otimes n} $, that is $P_i = \bigotimes_{i =1 }^n \sigma_i$ with $\sigma_i \in \{I,X,Y,Z\}$, where
$$
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \hspace{1cm} X =...
- 131
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
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 $(D,A)$ ...
- 4,058
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
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 ...
- 23
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
votes
1
answer
720
views
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 ...
- 287
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
answer
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\...
- 1,081
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 $\...
- 143
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 $i∈{1,…,n}$ ...
- 145
4
votes
1
answer
408
views
Proof of Levinson-Durbin algorithm
Is there any article or reference book with a full proof of the Levinson-Durbin algorithm used for solving linear system with a Toeplitz matrix ?
- 41
0
votes
0
answers
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 $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
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
1
vote
1
answer
737
views
How do the singular values of a Hankel matrix, generated by some data time series, change when we add/remove rows and columns?
Suppose I have a smooth time series $C(t)$ defined on the interval $t=[0,T]$, from which I extract the sub-series $c=\{x_1,\cdots,x_N\}$ of $N$ entries, where $x_i=C(i*T/N)$. Naturally, the number $N$ ...
- 11
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 ...
- 1,043
1
vote
1
answer
106
views
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: $A\in\mathbb{R}^{d\times d}$...
- 245
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
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 ...
- 723
0
votes
0
answers
46
views
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 ...
- 1
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 \...
- 55
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
$$D=\left[\begin{array}{cccc}
1^{- 2 p} & 0 & \cdots & 0 \\
0 & 2^{-2 p} & \cdots & 0 \\
\vdots &...
- 11
1
vote
1
answer
201
views
Matrix equation involving quadratic form
Let $X,Y\in\mathbb{R}^{n\times k}$, $\Lambda(\alpha) = \text{diag}(\alpha)$, with $\alpha\in\mathbb{R}^k$, and let $c,d\in\mathbb{R}^+$ be positive constants. Let
$$A_i(\alpha) = (X\Lambda(\alpha) X^...
- 245
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
$$
where $m \leq n$, and the ...
- 197
0
votes
0
answers
146
views
Square root of a circulant matrix block
I'm trying to show the following:
Given the following $n\times n$ symmetric circulant matrices
$$A^*=\begin{pmatrix}
1 & -\mu_a & 0 & ...&0&-\mu_a \\
-\mu_a & 1 & -\mu_a &...
0
votes
1
answer
74
views
State-dependent positive definite matrix
Consider a function $f(\mathbf{x})=\mathbf{M}_\mathbf{x}$ that outputs a nonsymmetric matrix $\mathbf{M}_\mathbf{x} \in \mathbb{R}^{N \times N}$ given an input vector $\mathbf{x} \in \mathbb{R}^N$.
Is ...
- 1
0
votes
0
answers
90
views
Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?
Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...
- 463
0
votes
0
answers
195
views
Proof negative-definiteness of a nonsymmetric and rank-deficient matrix
Consider the vectors $\mathbf{a} \in \mathbb{R}^N$ and $\mathbf{b} \in \mathbb{R}^N$ with $N>1$ and $\mathbf{a} \neq \mathbf{b}$.
The product $\mathbf{C}=\mathbf{a} \mathbf{b}^T \in \mathbb{R}^{N \...
- 1
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
0
votes
0
answers
141
views
Parseval's equivalent of Norm that includes a Projection matrix
I need to optimize the norm, ${\bf x}^H {\bf P}_{\bf B} {\bf x} $, where, ${\bf P}_{\bf B} = {\bf B}^H({\bf B} {\bf B}^H)^{-1} {\bf B}$, ${\bf B}$ is a known $M \times N$ matrix, with $M < N$ and $...
- 1
8
votes
1
answer
746
views
Counting eigenvalues without diagonalizing a matrix
Today's arXiv has a paper by Pierpaolo Vivo, Index of a matrix, complex logarithms, and multidimensional Fresnel integrals, which asks the question whether it is possible to calculate the number $N(\...
- 188k
1
vote
0
answers
139
views
A lower-bound on matrix-function with vector product
I am currently trying to show that a sequence of homeomorphisms converges to some limiting homeomorphism using Anderson's the inductive convergence criterion. However I can't explicitly compute the ...
- 5,405
0
votes
1
answer
533
views
Follow up: Show that these vectors are linearly independent almost surely
I posted this question some time ago here. I started a bounty for it and received an answer which helped me a lot. However, I still have some issues I want to discuss regarding it. Unfortunately I can'...
- 344
2
votes
1
answer
156
views
Characterize matrix range
$\DeclareMathOperator{\col}{\operatorname{col}}\DeclareMathOperator{\diag}{\operatorname{diag}}\DeclareMathOperator{\Range}{\operatorname{Range}}$Let $A \in \mathbb{R}^{n\times m}$, $D = \diag(d) = \...
- 245
11
votes
1
answer
1k
views
Show that these vectors are linearly independent almost surely
So I'm doing research in control theory and I have been stuck with this problem for a while. Let me explain my issue, then my proposal, and finally my concrete question.
Problem: I have $m<n$ real $...
- 344
0
votes
0
answers
149
views
L_q matrix inequality
The following arose out of studying $\ell_q$ Lewis weights. Let $P$ be a real $n \times n$ orthogonal projection matrix (i.e., $P$ is symmetric and $P^2 = P$) and let $W$ be the diagonal matrix ...
- 101
-1
votes
1
answer
172
views
$A\geq B\Rightarrow A^{-1}\leq B^{-1}$ entrywise for pos.def. symmetric matrices?
My question follows from https://math.stackexchange.com/questions/3857976/inverse-inequality-of-symmetric-matrix. Suppose we assume that $A$ and $B$ are two positive definite matrices with positive ...
- 154
2
votes
1
answer
448
views
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{...
- 61
2
votes
0
answers
588
views
Bounding Frobenius norm of pseudo-inverse
$\DeclareMathOperator{\F}{\mathrm{F}}$Let $\mathbf{A}$ and $\mathbf{A}^\prime$ be two $m\times n$ matrix such that $\|\mathbf{A}-\mathbf{A}^\prime\|_{\F}\leq \delta$. Is there any bound for the ...
- 287