Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,875 questions
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Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices
If $A$ is chosen uniformly at random over all possible $n$ by $n$ Toeplitz (or circulant) (0,1)-matrices, can we give any bounds for the expected size of the determinant of $AA^T$? All arithmetic is ...
- 3,377
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3
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Eigenvectors of a symmetric positive definite Toeplitz matrix
I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better.
Although I assumed this would be a well ...
- 111
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History of Jordan Canonical Form?
Can anyone suggest a reference that discusses the history of the Jordan canonical form? In particular, I am interested in:
When and how was it first stated? (I understand it was independently stated ...
- 3,761
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2
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Harmonic function properties on $\mathbb R^3$
Let $X$ be the set of all harmonic functions external to the unit sphere on $\mathbb R^3$ which vanish at infinity, so if $V \in X$, then $\nabla^2 V(\mathbf{r}) = 0$ on $\mathbb R^3 - S(2)$ and $\...
- 211
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Further aspects of a Hankel matrix of involution numbers
We have two conjectured generalizations of the question asked at
a Hankel matrix of involution numbers
by Tewodros Amdeberhan. Let $n!!=1!\,2!\cdots n!$.
Conjecture 1. Let $I_k$ denote the number of ...
- 50.8k
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4
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Probability two products are equal
I am interested in the following simple looking problem on which I am stuck. Let $M$ be a fixed $m$ by $n$ matrix with $\pm1$ elements. Let $x$ and $y$ be two independently sampled random $n$-...
- 3,377
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Is this generalization of eigenvalue and eigenvector studied?
While thinking about what it means for observables to be simultaneously measurable in quantum mechanics I came up with the following concepts, which I will call "linearly indexed" versions of standard ...
- 2,895
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Matrices whose nullspace is nicely shaped
I'm looking for natural conditions on $a_{ij}$ to guarantee that the null space of the $n\times m$ matrix $A=(a_{ij})$ has a nice basis.
The null space of { {1,-2,1,0,0}, {0,1,-2,1,0}, {0,0,1,-2,1} } ...
- 9,756
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2
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Existence of sparse LU decomposition of sparse matrix
Let $A$ be a sparse matrix over some field. I would like to know about the existence of LU decompositions so that $L,U$ are both sparse.
More precisely, let $A$ be an $N$-by-$N$ matrix. Suppose each ...
- 547
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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 $...
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Reference request: Volume 2 of Abhyankar's lectures on algebra?
Abhyankar has a magnificent, if meandering (check them out if you want to see what I mean), set of lectures on algebra.
The description:
This book is a timely survey of much of the algebra developed ...
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What is the order of the largest subset of M_n(Z_p) such that no two elements commute?
Let $A(n,p)$ be the order of the largest subset of $M_n(Z_p)$ such that no two distinct matrices in this subset commute. Is it true that $\lim_{p \to \infty} \dfrac{A(n,p)}{p^{n^2}} =1$? Can anyone ...
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How to determine the sign for the sum over all simple paths in the graph
$\DeclareMathOperator\perm{perm}\DeclareMathOperator\len{len}$Let $A$ be the adjacency matrix of a tree $T$ for some ordering $v_1,...,v_n$ of the vertices, and let $D=xI-A$ its characteristic ...
- 1,362
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Characterization of certain 4-dimensional lattices
Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$.
The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
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Matrices that admit a power that is symmetric
We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
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Fast computation of matrix product $AXA^T$ with fixed $A$?
Suppose we have two $n$-by-$n$ matrices $X$ and $A$, where $A$ is known and $X$ may change in different invocations, and we want to compute $AXA^T$. Is there an algorithm that beats the naive one of ...
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Jaffard's theorem - finite matrices
For infinite matrices, Jaffard's theorem states that if $(A(k,l))_{k,l\in \mathbb{Z}}$ is invertible and satisfies
$$
A(k,l) \leq C (1+\left|k-l\right|)^{-r},
$$
for some $C>0$,
then
$$
A^{-1}(k,...
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Formula or estimates for $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$
Given two positive definite matrices $A,B$ with nonnegative entries, I seek convenient ways to analytically compute or estimate $\frac{\operatorname{Tr}(AB)}{\operatorname{Tr}(A)}$, where $\...
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How the idea of adjugate matrix has been designed? [closed]
I can understand the adjugate matrix and the motivation of that to find the inverse, but I can't see how this idea was invented by mathematicians. It's just brilliance or someone understand how the ...
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An elementary linear algebra problem
Let $K$ be a field, and let $E$ be the algebra of $n\times n$ matrices over $K$. Let $V_0$ and $V_1$ be the (left) $E$-modules of matrices of size $n\times n_0$ and $n\times n_1$. Let $W \subseteq V_0$...
- 4,015
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Generalized Classical Adjoints and Factorizations of the Characteristic Polynomial
This is idle noodling, and I'm prepared to learn that it's foolish as well as idle. But....
Let $M$ be an $n\times n$ matrix over, oh, let's say an algebraically closed field for now. There have ...
- 23k
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Quantifying the failure of the Cholesky factorization test for indefinite matrices
The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff it is positive ...
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Proof that bases etc. exist in early linear algebra course?
I'm currently struggling to teach a 2nd course on linear algebra (in the UK, not at an Oxbridge quality university: the students have done a 1st course which concentrated upon algorithms you can apply ...
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Is $Sym^n (V^*) \cong Sym^n (V)^\ast$ naturally in positive characteristic?
Background/motivation
It is a classical fact that we have a natural isomorphism $Sym^n (V^*) \cong Sym^n (V) ^\ast$ for vector spaces $V$ over a field $k$ of characteristic 0. One way to see this is ...
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Determinantal symmetry: proof requested: Part I
Consider the determinantal function
$$F(a,b,c):=\det\left[\binom{i+j+a+b}{i+a}\right]_{i,j=0}^{c-1}.$$
I would like to ask:
QUESTION. Can you provide an argument, combinatorial or otherwise, to ...
- 43.2k
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Sum of Gaussian binomial coefficients.
We all know that $\sum_{i=0}^{n}{n \choose i}=2^{n}$. Is there a similar result regarding the q-binomial coefficients? (a.k.a Gaussian binomial coefficients) - $\sum_{i=0}^{n}{n \choose i}_{q}=?$
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Direct product of free groups in $\mathrm{SL}_3(\mathbb{Z})$
Let $\mathbb{F}_2$ be the free group on two generators. Does $\mathbb{F}_2 \times \mathbb{F}_2$ embed as a subgroup of $\mathrm{SL}_3(\mathbb{Z})$?
- 1,769
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Find the inverse of a matrix that is very similar to the Hilbert matrix
The standard Hilbert matrix $H$ is given by
$$H_{ij}=\frac{1}{i+j-1},$$
and it has an inverse given for example in this MO question.
Now I have encountered a matrix $M$ of similar form, namely,
$$...
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Shifted QR algorithm—why does the shift help?
I read that a way to speed up the convergence rate of the QR algorithm is to shift the target
matrix. It is not so clear to me why this helps. The convergence rate depends on the
minimum gap between ...
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Any reference on multilinear algebra [closed]
Do you know any good reference on multilinear algebra?
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Partial inverse of a matrix - or does it have its own name?
In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name.
That is, a matrix (here ...
- 1,612
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Derivative of a determinant of a matrix field
Let $A(x_1,...,x_n)$ be an $n\times n$ matrix field over $R^n$.
I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that:
$\frac{\partial{\det(A)}}{\...
- 995
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Bounding the trace of a matrix product by the operator norms; generalized Hölder inequality?
$\DeclareMathOperator\Tr{Tr}$Let $A_i$ with $i=1,\dotsc,N$ and $p$ be real $M\times M$ matrices.
Further, let $p$ be positive definite, i.e., $p\succ 0$, with $\Tr(p)=1$. Let $0< a_i<1$ and $\...
- 103
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Non-conjugate words with the same trace
Let n>=2, p a large prime, G = SL_n(Z/pZ).
If n=2, there are words that, while not conjugate in the free group, do have identical trace in G. For example, tr(g h^2 g^2 h)= tr(g^2 h^2 g h) for all g, ...
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Formal power series is Taylor expansion of rational function iff Hankel determinants vanish?
Let $$ u(T)=\sum_{n = 0}^\infty a_nT^n$$ be a formal power series over a field $K$. Then why does $u(T)$ lie in $K(T)$ (i.e. is the Taylor expansion of a rational function) if and only if there is an $...
user97565
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Maximum rotation made by a symmetric positive definite matrix?
Say we have some symmetric positive definite $n\times n$ matrix $M$ with $n$ distinct eigenvalues $\{\lambda_1,...,\lambda_n\}$. Is there a general formula for the maximum angle $\theta$ for which $M$ ...
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Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector
In a lot of problems in linear algebra one uses the existence, for each $E$ vector space over a field $k$, and each $x\in E$, of a Hyperplane $H$ such that $E=k\cdot x \oplus H$ (Let us denote $\...
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Relationship between free probability and deterministic graphs?
Consider the $N\times N$ matrix $$
M = \left(\begin{array} \\
0 & 1 & & 0 \\
1 & \ddots & \ddots & \\
& \ddots & \ddots & 1 \\
0 & & 1 & 0 \\
\end{...
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When does $\det(\frac{A+A^T}{2})=\det(A)$ for positive-definite $\frac{A+A^T}{2}$?
Setup: Let $A$ be a real square matrix and assume its symmetric part $\frac{A+A^T}{2}$ is positive-definite. The inequality
$$
\det\left(\frac{A+A^T}{2}\right) \leq \lvert\det(A)\rvert
$$
is known as ...
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How do you tell if a system of linear inequalities has a solution?
A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned. I imagine there must be a more direct way.
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Determinant of the "real part" of a matrix
Let $A$ be an $n\times n$ complex matrix, and write $A=X+iY$, where $X$ and $Y$ are real $n\times n$ matricies. Suppose that for every square submatrix $S$ of $A$, $|\mathrm{det}(S)|\leq 1$ (i.e., ...
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Coefficient-wise powers of matrices. Reference wanted
Let $K$ be a commutative field and ${\rm M}_n (K)$ be the ring of $n\times n$ square matrices with coefficients in $K$ ($n\geqslant 1$ is an integer). For $k\geqslant 1$ and $A =(a_{ij})_{1\leqslant i,...
- 6,349
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Growth of an integer vector under the action of a matrix in $GL_n(\mathbb{Z})$
I have some questions regarding the dynamics of elements of $GL_n(\mathbb{Z})$ acting on $\mathbb{Z}^n$. In particular, given an invertible integer matrix $M \in GL_n(\mathbb{Z})$, and given an ...
- 15.4k
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A strange matrix equality
Let $A$ and $B$ be $n\times n$ real matrices.
When $n=2$, we have the equality
$$A\Big(\mbox{Trace}(B)A-\mbox{Trace}(A)B\Big) B=B\Big(\mbox{Trace}(B)A-\mbox{Trace}(A)B\Big) A.$$
Can we give an ...
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4
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How to solve Ax=b incrementally ?
Hi, everyone.
What I am struggling is the following problem. I have a linear matrix equation $Ax=b$, where $A$ is a known $n \times n$ large sparse real matrix, $x$ and $b$ are known $n \times 1$ ...
- 101
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How many Lie and associative algebras over a finite field are there?
This question is related to the following general question:
Given a variety of (non-associative) algebras $\mathcal V$, a finite field $\mathbb{F}_q$, with $q$ elements, and a positive integer $n$, ...
- 398
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Distinguishing combinatorial maps by their linearizations
Every (not-necessarily invertible) map $f$ from $[n]:=\{1,2,,,,.n\}$ to itself determines a linear map $L_f$ from ${\bf R}^n$ to itself that sends the basis vector $e_k$ to $e_{f(k)}$ for $1 \leq k \...
- 19.7k
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$2 \times 2$ matrix question
Let $A$, $B$, and $C$ be $2\times 2$ complex matrices, with $A$ and $C$ rank $1$ Hermitian. Can we find a real number $a$ and a $2\times 2$ unitary $U$ such that
$$A + BV + V^*B^* + V^*CV$$
is a ...
- 42.8k
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Characteristic polynomial of exterior power
Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of ...
- 96.4k
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Perturbation theory for the generalized eigenvalue problem
Is there a standard reference for the perturbation theory of the generalized eigenvalue problem?
More specifically, I would like to get a systematic expansion for the problem
$(A_0 + \epsilon A_1)...
- 1,193