Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,073
questions
0
votes
0
answers
15
views
Proof that sum of k-eigenvalues is convex
I saw a post A sum of eigenvalues that said
It is well-known that $\sum^{r}_{i = 1} \lambda_{i}(X)$ is convex.
and I saw an explanation in Boyd and Vandenberghe - "Convex Optimization", ...
-4
votes
0
answers
27
views
Solving a 2x2 system of linear equations that is inconsistent or consistent dependent [closed]
For each system, choose the best description of its solution.
If applicable, give the solution.
-5x+y=-5 and 5x=5+y
The system has no solution.
The system has a unique solution:
The system has ...
3
votes
1
answer
105
views
A system of $2N$ equations resembling a Vandermonde matrix
Suppose that I have a collection of known $\gamma_1, \dots, \gamma_{2N} \in \mathbb{C}$. Is there a known method to compute $\zeta_1, \dots, \zeta_N, \alpha_1, \dots, \alpha_N \in \mathbb{C}$ that ...
- 475
1
vote
0
answers
39
views
Are these $L_2$-spectral radii approximations strictly increasing?
Suppose that $V$ is a finite dimensional complex Hilbert space. Let $L(V)$ denote the collection of all linear mappings from $V$ to $V$. Let $A_1,\dots,A_r:V\rightarrow V$ be linear operators. Then ...
- 22.4k
1
vote
0
answers
47
views
Integer programming using the Steinitz lemma
I am trying to implement an algorithm that I read on the paper entitled: "Proximity results and faster algorithms for integer programming using the Steinitz lemma", published by Friedrich ...
3
votes
0
answers
32
views
Invertibility of the sampling matrix
Given a function $f: \mathbb{R}^2\rightarrow\mathbb{C}$ sampled as a matrix $F_{ij}$ on some ractangle $[a,b]\times[c,d]\subset\mathbb{R}^2$ with steps $\Delta x$ and $\Delta y$ as the stepsizes so ...
- 71
3
votes
0
answers
77
views
Construct a special kind of SVD
Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as
$$ A = XD_AY^H \\
B = XD_BY^T $$
where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
- 71
1
vote
0
answers
80
views
Construct special "joint SVD" from separate SVDs
Given two matrices, $A,B\in\mathbb{C}^{n\times n}$ which can be written as
$$ A = XD_AY^T \\
B = XD_BY^T $$
where $X$ and $Y\in\mathbb{C}^{n\times n}$ are unitary and with diagonal $D_A$ and $D_B\in\...
- 71
3
votes
0
answers
35
views
Derivative of characteristic polynomial of a graph and derivative of characteristic polynomial of a vertex-deleted subgraph have a common root
Let $G$ be a simple graph and $G-i$ be one of its vertex-deleted subgraphs. Let $\phi(G,x)$ and $\phi(G-i,x)$ be the characteristic polynomials of $G$ and of $G-i$ respectively, with respect to their ...
- 317
2
votes
0
answers
48
views
Can the supremum of this quotient of spectral radii be reached?
Let $V$ be a finite dimensional complex inner product space. If $A_1,\dots,A_r\in L(V)$, then define a mapping $\Phi(A_1,\dots,A_r):L(V)\rightarrow L(V)$ by letting $\Phi(A_1,\dots,A_r)(X)=A_1XA_1^*+\...
- 22.4k
1
vote
0
answers
37
views
Find $\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right)$
I am doing a quantum optimization where the final problem has the following form
$$\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right),$$
where $V \in \mathbb{C}^{d\times ...
- 11
1
vote
0
answers
49
views
A question about an inequality for hafnians of some special matrices
Let $S$ by a complex symmetric $2m$ by $2m$ matrix. Let $\sigma$ be the $2m$ by $2m$ matrix which is the direct sum of $m$ copies of the following matrix:
$$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \...
- 3,594
0
votes
0
answers
57
views
Submodule of matrix space is free
Let $\mathcal{H}$ be an infinite dimensional complex inner product space, and denote by $\mathcal{H}^{n \times n} = \mathcal{H} \otimes_{\mathbb{C}} \mathbb{C}^{n \times n}$ the corresponding matrix ...
2
votes
1
answer
87
views
Probability density of a hyperplane for a Gaussian distribution
I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution
$$P[\textbf{x}\in S]
=\int_{\textbf{x}\in S}
\det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
- 162
0
votes
0
answers
25
views
What are the convergence requirements for Inverse Power Method?
I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below.
...
1
vote
0
answers
49
views
Eigenvalues of two positive-definite Toeplitz matrices
Consider two positive-definite Toeplitz matrices $M_1$ and $M_2$ both with dimension $2^j \times 2^j$. Their matrix elements are:
$$M_1[x,y] = \frac{\text{sin}(\pi(x-y)/2^j)}{\pi(x-y)} \qquad M_2[x,y] ...
- 11
0
votes
0
answers
53
views
Determining the total number of nonzero expansion terms in a (0,1)-matrix
Let $A=(a_{ij})_{n\times n}$ be a $(0,1)$-matrix such that it contains equal number of $1$s in each row and column. Is there any general method to count the total number of the nonzero terms $\prod_{i=...
- 133
1
vote
0
answers
84
views
$r(M)$-subsets of a 3-connected matroid $M$
It is proved in Lowrance, Oxley, Semple, and Welsh - On properties of almost all matroids that almost all matroids are 3-connected asymptotically. Also, it is conjectured that almost all matroids are ...
- 359
0
votes
0
answers
134
views
CRT for linear forms
Suppose $A,B,C,A',B',C'$ are random distinct primes in $[T,2T]$ and $u,v$ are integers in $[T,2T]$.
Suppose we know:
$$Au+Bv\equiv r\bmod C$$
$$A'u+B'v\equiv r'\bmod C'$$
can we identify $u,v$ in ...
- 12.8k
3
votes
2
answers
139
views
Power of a matrix, largest eigenvalue in absolute value, and convergence acceleration
I want $S^k$, with $S=I-\Lambda^{-1}M$, to tend to zero quite fast as $k\rightarrow \infty$, as this is what drives the convergence in a fixed-point algorithm. Here $M=X^TX$ is a fixed $m\times m$ ...
- 2,279
0
votes
0
answers
31
views
When is the Fourier discrete matrix almost similar to some particular diagonal matrix?
Let $N$ be a natural number and put $z=e^{\frac{\pi i}{N}}$ and $w=z^2$. Let us consider the discrete Fourier matrix $F=(w^{kl})_{k,l=0,\cdots,N-1}$ and the diagonal matrix $D=\operatorname{diag}(1,...
- 3,343
1
vote
0
answers
68
views
A conjecture about comparing determinants of two Toeplitz matrices
Let $p$ be an odd prime number and denote $m=(p-1)/2$. Let $\mathbb F_p$ be the finite field of order $p$. My question is about comparing determinants of two matrices over the polynomial ring $\mathbb ...
- 323
0
votes
0
answers
75
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 (...
1
vote
1
answer
37
views
Lipschitz continuity and quadratic growth in Loewner order
Consider the partial Loewner order $\le_L$ for symmetric matrices: let $A,B$ be symmetric matrices of the same dimension, we say $A\le_L B$ if $B-A$ is positive definite. Now let $f:\mathbb{R}^n\to \...
- 423
0
votes
1
answer
112
views
Semi simplicity over commutative algebras over non-algebraically closed fields
I have already posted this on stackexchange
I have a question:
If k is an arbitrary field then is it true that if $M$ a finite dimensional $k[x, y]$ is semisimple as a $k[x]$ module and also as a $...
- 121
1
vote
1
answer
138
views
History of the characteristic matrix
Let $\mathbb{F}$ be a number field, $A$ and $B$ be two $n\times n$-matrix over $\mathbb{F}$. It is known from some textbook that $A$ is similar to $B$ iff there exists $n\times n$ nonsingular matrix $...
- 571
5
votes
1
answer
160
views
Similarity of a matrix with its transpose
An $n\times n$ matrix $A$ over a number field is similar to its transpose $A^T$. Is there any natural way to construct a nonsingular matrix $P$ such that $P^{-1}AP=A^T$?
- 571
3
votes
0
answers
157
views
Product of Hermitian forms over a group ring
Let $G$ be a group and $k=Z[G]$ the corresponding group ring equipped with the standard involution.
Let $(P,p)$ and $(Q,q)$ be two $\epsilon$-quadratic forms, $\epsilon = \pm1$, where $P$ and $Q$ are ...
- 43
1
vote
0
answers
78
views
When does the Cauchy-Schwarz inequality for spectral radii of tensor products become equality?
Let $V$ be a complex finite dimensional inner product space. If $A_{1},\dots,A_{n}:V\rightarrow V$ are linear operators, then let $\Phi(A_{1},\dots,A_{n}):L(V)\rightarrow L(V)$ be the superoperator ...
- 22.4k
0
votes
0
answers
73
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 761
0
votes
0
answers
21
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
0
votes
0
answers
36
views
Determinant of barycenter of a hyperbolic-matrix
Let $A \in \operatorname{GL}(d, \mathbb{R})$ be a hyperbolic matrix. I want to show that $$\det((1-\alpha)A+\alpha\operatorname{Id})\geq 1,$$
where $0<\alpha<1$.
Attempt:
In $\operatorname{SL}(2,...
- 761
1
vote
1
answer
32
views
(Approximation) Algorithms for Weight Distribution / Subspace Weights Problem in coding theory
The Weight Distribution / Subspace Weights Problem in coding theory is defined as this:
Instance: A binary $m$ by$n$ matrix $H$ and an integer $k > 0$
Question: Is there a set of $k$ columns of $...
- 31
2
votes
1
answer
93
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 $(D,A)$ ...
- 3,343
0
votes
0
answers
28
views
Integration of matrix form of Vasicek variance (Python/Matlab)
$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
- 1
1
vote
0
answers
70
views
On the generation of linear groups
$\def\GL{\operatorname{GL}}\def\id{\mathrm{id}}$Let $V$ be a finite dimensional vector space over a (commutative) field $k$. If $f:V\to V$ is a linear map I'll write $r_f$ and $V^f$ for the rank of ...
- 45.1k
2
votes
0
answers
38
views
spilt the sum of singular values of matrices
Let $A_{i} \in GL(d, \mathbb{R})$ for $i=1, 2, 3.$ For $q>0$, we denote $t_{3}^{q}=\sum_{i=0}^{3} \sigma_{1}^{q}(A_{i})\sigma_{2}^{q}(A_{i})\sigma_{3}^{q}(A_{i})$, $t_{2}^{q}=\sum_{i=0}^{3} \sigma_{...
- 761
0
votes
1
answer
138
views
Derivative involving a singular matrix
Find the first derivative of a SUM of all elements of an INVERSE of a square matrix (whose elements are functions of $z$) at $z=1$ knowing that all of the matrix' elements evaluate to $1$ at $z=1$.
...
- 409
1
vote
0
answers
59
views
Proving the non-existence of canonical isomorphisms
From time to time, during my undergraduate lectures on linear algebra appears the following question from the most smart students in the class. I asked to my algebra colleagues but I have not received ...
- 1,054
1
vote
0
answers
63
views
Find the eigenvectors from the QR algorithm in the unsymmetric case
It is possible to find many references describing the QR Algorithm with more or less refinements to approximate the eigenvalues of a square matrix $A\in\mathbb{R}^{n\times n}$.
I implemented a version ...
- 11
5
votes
0
answers
66
views
Isomorphism between tensor product of exterior power spaces
Suppose that $V_1, V_2, V_3$ are finite dimensional vector spaces over $\mathbb{C}$ of dimensions $d_1, d_2, d_3$, respectively. Suppose that $V_1, V_2, V_3$ are equipped with inner products, so that ...
- 650
1
vote
0
answers
45
views
Tucker decompositions over arbitrary fields
Given an $n$-mode tensor $\mathcal{T}\in\mathbb{R}^{d_1\times\dotsb\times d_n}$, there exists a Tucker decomposition of $\mathcal T$ of the form
$$\mathcal{T} = \mathcal{X}\times_1 W_1\times_2\dotsb\...
- 228
1
vote
0
answers
26
views
Complexity of singular value decomposition using matrix multiplication oracles
Suppose I have an $n\times m$ real matrix $A$, $n\ll m$ with full row rank $(\mathrm{rank}(A) = n)$. I have an oracle that can compute $Ax$ or $A^T y$ for any $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
- 363
2
votes
1
answer
86
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\...
- 423
0
votes
1
answer
51
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
1
vote
0
answers
77
views
A dimension problem related to an abelian simple extension of a field
$\DeclareMathOperator\Imm{Im}$Let $K=F(\alpha)$ be an abelian extension of $F$ and let $\sigma$ be a map (could be any map) from $K^\times$ (the multiplicative group of $K$) to itself. Define an $F$-...
- 213
0
votes
1
answer
83
views
Proving maximum value of a determinant of $I - B$, where $B$ is nonnegative matrix
I have the following setting:
Let $0 \leq r < 1$ and let $\{z_i\}_{i=1}^k$ be $k$ complex numbers such that $|z_i| \leq r$ for all $i$.
Moreover, $r + \sum_{i=1}^k 2Re(z_i) \geq 0$
I am interested ...
- 59
1
vote
1
answer
264
views
A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
- 3,343
14
votes
3
answers
1k
views
Why is the set of Hermitian matrices with repeated eigenvalue of measure zero?
The Hermitian matrices form a real vector space where we have a Lebesgue measure. In the set of Hermitian matrices with Lebesgue measure, how does it follow that the set of Hermitian matrices with ...
- 275
3
votes
2
answers
109
views
Iterative methods for linear system with non-diagonally dominant matrix
I have a linear system
\begin{align*}
\left[\begin{array}{cccc}
1 & 2 & 1 & -1 \\
3 & 2 & 4 & 4 \\
4 & 4 & 3 & 4 \\
2 & 0 &...
- 65