Questions tagged [linear-algebra]

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

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Recovering Shared Eigenvector Set

Suppose we are given a set of $M$ pairs $\{(\vec{x}^{(i)},\vec{y}^{(i)})\}$, with $\vec{x}^{(i)}\in\mathbb{R}^N$, $\vec{y}^{(i)}\in\mathbb{R}^N$, $M\gg N$ such that $\vec{y}^{(i)} = Q^{(i)} \vec{x}^...
MWB's user avatar
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Reduction of size of orthogonal matrices.

While experimenting with orthogonal vectors I've noticed the following transformation: If $$ A = \begin{bmatrix}z & r \cr c & B\end{bmatrix} $$ is orthogonal, $z$ ...
Bob Terrell's user avatar
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Intersection of subspaces

If you have two linear subspaces $V_1$ and $V_2$ of a vector space $V,$ both given by their bases, there is fairly heavy handed way of computing their intersection: write down the projection matrices ...
Igor Rivin's user avatar
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Large scale sparse system of linear equations

What is the best know algorithm for solving a large sparse system of linear equations? The system I'm working on is not symmetric, not positive definite and integer. The only benefit is being sparse. ...
Star's user avatar
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connectivity in automata by words of length n-1

Let $A$ be a complete strongly connected automaton with $n$ states. Does always exist a word $v$ of length at most $n-1$ such that its underlying graph is connected? That is for any pair of distinct ...
Mikhail Berlinkov's user avatar
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Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere. Let $A$ be an $l\times n$ matrix with ...
chris seaton's user avatar
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Algorithm/denominators of elements of a rational affine space

I hope it's not a trivial question... Suppose I have a finite dimensional vector space $V$ over $\mathbb{Q}$ with a distinguished basis (in my case it's the $k$th graded piece of the free associative ...
Adrien's user avatar
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Pseudoinverse of column submatrix, from pseudoinverse of entire matrix.

Hello, I am working on a numerical method for the least-squares solution of a linear system. I know that I can approximate the solution to $Ax=b$ with $x=A^+b$, where $A^+$ is the Moore-Penrose ...
Jason DeVita's user avatar
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The determinant of the hadamard product of two matrices

We know that the determinant of a Hadamard product of two positive semidefinite matrices $|{\bf A}\circ{\bf B}|$ is greater than or equal to $|{\bf A}||{\bf B}|$. Are there any general results on ...
Anadim's user avatar
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Field of Values and Eigenvalues

Let $A$ be an $n \times n$ matrix. Define the field of values of $A$, denoted $W(A)$, as $ W(A) := \{c \in \mathbb{C} : \exists x \in \mathbb{C}^n, \|x\|_2 = 1, x^H Ax = c \} $ The question is, ...
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Special linear group of a quotient

Let J be a non-trivial ideal of a commutative ring A. The canonical map from A to the quotient A/J induces a homomorphism $\varphi : SL_n(A) \to SL_n(A/J)$. In general $\varphi$ is not surjective (for ...
MathOMan's user avatar
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Convergence of the relaxation method for every parameter in the relevant disk

For large size matrices, the resolution of linear systems $Ax=b$ is often done iteratively. The matrix $A$ is split as $A=M-N$, with $M$ invertible, and one performs $$x^{k+1}=M^{-1}(Nx^k+b).$$ The ...
Denis Serre's user avatar
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Is it possible to use linear programming to solve this problem?

I am trying to write software to minimize pricing for cell phone subscription services, ie: choose the optimum plan for each customer in a large group. Could someone comment on whether this is ...
user6546's user avatar
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"Cholesky decomposition" X=YY* for p-adic matrices?

Let $E/F$ be a quadratic extension of $p$-adic fields. Consider $M_n(E)$ with the unitary (aka 2nd kind) involution $X \mapsto \sigma(X)^{tr}$, where $\sigma(X)$ denotes the entry-wise application of $...
Sean Rostami's user avatar
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bounded homogeneous quartics

If Q is a real homogeneous quartic on $R^N$, $Q(x) = \sum_{1 <= i,j,k,l <= N} Q_{ijkl} x_i x_j x_k x_l$ what is the condition on the (totally symmetric) coefficients $Q_{ijkl}$ for Q ...
Daniel's user avatar
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About a circular variant of Vandermonde matrix

Given an arbitrary $(x_1, \dots, x_n) \in [0, 1]^n$, is there any name/known results for the following $n \times n$ matrix (which is constructed by iterating $(x_1 \to \dots \to x_n \to x_1 \to \dots)$...
lntk's user avatar
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A stochastic matrix $B = \lambda(\lambda I - A)^{-1}$ such that $B-B^2$ has a non-negative diagonal

I apologize if this is too elementary a question, but I have not been able to make much progress. Consider a real matrix $A$ with $A_{ij} >0$ for $i \ne j$ and $\sum_{j} A_{ij} = 0$ for each $j$. ...
user133281's user avatar
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An isomorphic classification of non-associative division octonion algebras

A division octonion algebra over a field $F$ is a $8$-dimensional unital non-associative algebra $A$ over the field $F$, endowed with a quadratic form $N:A\times A\to F$ such that $N(xy)=N(x)N(y)$ and ...
Taras Banakh's user avatar
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Admissibility of Ulm's invariants

Let $G$ be a reduced abelian $p$-group. We set $G_0=G$. Let $\alpha$ be an ordinal. Inductively, if $\alpha=\beta+1$ is a successor ordinal, we define $$G_{\alpha}=pG_{\beta}.$$ If $\alpha$ is a limit ...
Nini's user avatar
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Metrizing pointwise convergence of *sequences* of functionals in a dual space

This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here: Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
Mustafa Motiwala's user avatar
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Stability of indefinitely damped mechanical system with diagonal stiffness

I'm trying to find conditions for the asymptotic stability of the following linear system, \begin{equation} \mathbf{I \ddot{x}} + \mathbf{B \dot{x}} + \mathbf{K x} = 0 \end{equation} given the ...
Shivang Rawat's user avatar
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185 views

Exponential growth of shortest vector norm for successive lattices corresponding to powers of a matrix

Let $A\in M_{2\times 2}(\mathbb{Z}) $ be a two by two integer matrix such that $0,\pm 1$ are not eigenvalues of $A$ and $\left|\det(A)\right|>1$. I am interested in the growth of the norm shortest ...
an_ordinary_mathematician's user avatar
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Solving an underdetermined system of linear equations among the rationals, in the neighbourhood of an approximate solution

I have a system of linear equations $Ax=b$. Extremely underdetermined, for concreteness $x \in \mathbb{R}^{17,000}, b \in \mathbb{Z}^{156}$. $A$ is sparse, integer, full rank. I have a very precise ...
Dániel Varga's user avatar
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When can the eigenvalues be constrained to the right half plane as a result of column permutations?

Suppose we have a real square matrix. Under what conditions is it possible to permute the columns of the matrix such that all eigenvalues of the resulting matrix have nonnegative real part?
Max Aifer's user avatar
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Computing Grothendieck group of coherent sheaves of affine toric 3-fold from a simplicial cone

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma$ be a 3-dimensional simplicial cone in $\mathbb{R}^3$. Let $X=\operatorname{Spec}(k[\sigma^{\vee}\cap\mathbb{Z}^3])$ be the ...
Boris's user avatar
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Unitary operators with the same inner product as vectors

Suppose we have a set of real unit vectors $v_1,\ldots,v_m \in \mathbb{R}^n$. We can always find a set of unitary operators $U_1,\ldots,U_m$ acting on $\mathbb{C}^N$ (for $N$ that is possibly much ...
user3826143's user avatar
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Which invertible linear maps preserve the set of Hurwitz stable matrices?

Let $V = M_n(\mathbb{R})$ be the set of all $n\times n$ matrices with real elements and $V_{-}$ be a subset of Hurwitz stable matrices, i.e. matrices such that all their eigenvalues have strictly ...
Fiktor's user avatar
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How many local maxima can $(x_1,\dots,x_r)\mapsto\|x_1A_1+\dots+x_rA_r\|_\infty/\|(x_1,\dots,x_r)\|_2$ have for Hermitian $A_1,\dots,A_r$?

Let $K\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}$. Suppose that $A_1,\dots,A_r\in M_n(K)$ are all Hermitian. Define a function $f_{A_1,\dots,A_r}:\mathbb{RP}^{n-1}\rightarrow[0,\infty)$ by setting $$f_{...
Joseph Van Name's user avatar
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Automorphisms of matrix algebras and Picard group

This is a repost of https://math.stackexchange.com/questions/4692364/automorphisms-of-matrix-algebras-and-picard-group (asked on MSE). Notation. In what follows, $R$ is a commutative ring with $1$, $n\...
GreginGre's user avatar
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Describing the outer automorphism of a special unitary group in terms of the Hermitian form

$\DeclareMathOperator\U{U}\DeclareMathOperator\SU{SU}\DeclareMathOperator\GL{GL}$Let $h$ be a non-degenerate Hermitian form on $\mathbb{C}^n$ with signature $(p,q)$. Let $\U_h$ denote the associated ...
stupid_question_bot's user avatar
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Efficiently finding solutions to the Rainbow cryptosystem using quotient spaces

Repost of a mathematics stackexchange question here as this concerns my research and it went unanswered on there. In this paper, Ward Beullens gives another way to look at the Rainbow cryptosystem. In ...
Saegusa's user avatar
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Dimension of a kernel of a cocycle map

Inspired by a previous question (Dimension of a kernel of a linear map) and some of the answers I was given I thought wheter I can generalize the question to the following: Compute the kernel (or at ...
Marcos's user avatar
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Efficient way to calculate Smith Normal Form of large integer matrices

I am interested in calculating the Smith Normal Form for Laplacian matrices of hypercube graphs. Using the elementary divisors method from SAGE, I was able calculate up to the 11-cube (which has a $2^{...
presidentediniente's user avatar
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206 views

efficient numerical algorithm for matrix determinant

It appears that in numerical analysis the question of computing the determinant $\det A$ of a real or a complex $n\times n$ matrix $A$ is not well-studied, and a usual recommendation is to use matrix ...
Dima Pasechnik's user avatar
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82 views

Exterior powers of the Cartan matrix and Dyck paths

(This question can be formulated purely combinatorially in terms of Dyck paths, which is done in the second part of the question. But I am more interested whether this can be explained by some sort of ...
Mare's user avatar
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Grothendieck schemes and the Sheffer differential op calculus (Rota, Roman, et al. finite operator calculus)

In "Left differential operators on non-commutative algebras" on p. 4, Michiel Hazewinkel displays "precisely the right definition of differential operator" as $$D\; X^n = F(\tfrac{...
Tom Copeland's user avatar
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On a paper of von Neumann

Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality $$ \lVert p(T)\rVert \leq \sup \...
HaSa's user avatar
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Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?

Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form \begin{equation} \begin{pmatrix} 0 & \Delta \\ -\Delta & 0 \end{pmatrix}, \quad ...
Carlos Esparza's user avatar
3 votes
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107 views

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 $...
ABB's user avatar
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3 votes
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292 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
Puzzled's user avatar
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140 views

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 ...
wlad's user avatar
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A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix $$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$ Let us denote by $s_{-,l}$ the $l^{\text{...
ABB's user avatar
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172 views

Where could a paper on a unification of matrix decompositions be published?

I've got a paper which shows that when the spectral theorem (as a statement that every self-adjoint matrix can be unitarily diagonalised) is naively generalised to $*$-algebras other than the complex ...
wlad's user avatar
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Checking the generic rank of a matrix

Suppose that $A,B\in M_{p,q}(\mathbb{Z})$ are two rectangular integer matrices of the same size. Suppose that one has a conjecture stating that the rank of the matrix $A+tB$ for Zariski generic values ...
Vladimir Dotsenko's user avatar
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0 answers
133 views

Unitary equivalence of symmetric and homogenous polynomials

Given any two symmetric and homogenous polynomials with complex coefficients, I'm trying to determine if a unitary change of basis relates them. Specifically, assuming the polynomials are of degree $n$...
Deepesh Singh's user avatar
3 votes
0 answers
565 views

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 $M$ ...
getraparth's user avatar
3 votes
0 answers
44 views

Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size

I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size). This is equivalent to $$\...
CWC's user avatar
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"Circulant-Vandermonde" matrix: in search of a formula

An $n\times n$ circulant matrix $\mathbf{X}_n$ has the form \begin{align} \mathbf{X}_n= \begin{bmatrix} x_1 & x_2 & \cdots & x_{n-1} & x_n \\ x_2 & x_3 & \cdots & x_n&...
T. Amdeberhan's user avatar
3 votes
0 answers
116 views

Preconditioners for $Ax=y$ that rely on hierarchical statistical modeling

Solving $Ax=y$ exactly can be done as: fit a linear autoregressive model by treating rows of $A$ as data apply this model to $A^T y$ Imperfect predictive model corresponds to an approximate inverse ...
Yaroslav Bulatov's user avatar
3 votes
0 answers
70 views

Is the Kalman Filter computationally optimal for Kalman filtering?

Kalman filtering is known to be a recursive process that minimizes mean square error in linear problems. My question is: has anybody shown that this algorithm is computationally optimal, i.e. that you ...
Diego Méndez's user avatar

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