Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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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", ...
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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 ...
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3 votes
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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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\...
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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 ...
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2 votes
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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^*+\...
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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 ...
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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 \...
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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 ...
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2 votes
1 answer
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Probability density of a hyperplane for a Gaussian distribution

I have a vector $\mathbf{x}$ with a multivariate Gaussian distribution $$P[\textbf{x}\in S] =\int_{\textbf{x}\in S} \det(2\pi H^{-1})^{-1/2}\exp(-\frac{1}{2} \textbf{x}^T H\textbf{x}) \, d\textbf{x}$$...
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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. ...
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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] ...
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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 $\prod_{i=...
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$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 ...
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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 ...
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3 votes
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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 $m\times m$ ...
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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,...
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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 ...
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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 (...
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1 vote
1 answer
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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 \...
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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 $...
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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 $...
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5 votes
1 answer
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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 $P^{-1}AP=A^T$?
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3 votes
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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 ...
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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 ...
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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 ...
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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 ...
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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,...
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1 vote
1 answer
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(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 $...
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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 $(D,A)$ ...
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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}\...
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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 ...
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2 votes
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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_{...
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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 $z=1$. ...
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  • 409
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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 ...
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1 vote
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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 ...
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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 ...
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  • 650
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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\...
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1 vote
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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 $x\in \mathbb{R}^m, y\in \mathbb{R}^n$. ...
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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\...
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  • 423
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1 answer
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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 ...
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1 vote
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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 $F$-...
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0 votes
1 answer
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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 ...
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  • 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 $F$ ...
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  • 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 ...
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  • 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 &...
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