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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Local-globalism for similar matrices?
My background on number theory is very weak, so please bear with me...
Given two matrices $A$ and $B$ in $\mathbb{Z}^{n\times n}$. Assume that for every prime $p$, the images of $A$ and $B$ in $\...
- 33.8k
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550
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Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?
Let $a_1, \dots, a_n$ be a finite set of positive reals. Is there a $\mathbb Q$-basis of $\mathbb R$ where each $a_i$ has nonnegative coordinates?
Playing around with the case $n = 2$, I’m pretty sure ...
- 64k
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Canonic identification of the tangent space of the Grassmannian
let $Gr(k,V)$ be the grassmannian of k-dimensional subspaces of the complex vector space $V$ of dimension $n>k$.
I know that, given $K\in Gr(k,V)$, $T_{Gr(k,V),K}\simeq Hom(K,V/K)$, but i want to ...
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Is the eigenvalue map open?
The eigenvalue map in question is
$\sigma: {\mathfrak gl}(\mathbb{C}, n) \to S_n \backslash \mathbb{C}^n$,
from $n$ by $n$ complex matrices to $\mathbb{C}^n$ vectors modulo permutation of entries by $...
- 1,123
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520
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Problems concerning subspaces of $M_{n}(\mathbb{Q}) $
Let $M_{n}(\mathbb{Q}) $ denote the $n$ times $n$ matrices over the rational number field. $N$ be a subspace of $M_{n}(\mathbb{Q}) $.Then if all the non-zero matrices in $N$ are invertible, what is ...
- 923
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How can we make precise the notion that a finite-dimensional vector space is not canonically isomorphic to its dual via category theory?
There are quite a few questions both on this site and math.SE related to this topic as well as what we mean when we say "natural" or "canonical". For the purposes of this question, ...
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A binomial determinant fomula
Is there an existing or elementary proof of the determinant identity
$
\det_{1\le i,j\le n}\left( \binom{i}{2j}+ \binom{-i}{2j}\right)=1
$?
- 171
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1
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(Barely) linearly independent vectors over $\mathbb{Z}/2\mathbb{Z}$
Let $V$ be a vector space over $\mathbb{Z}/2\mathbb{Z}$. Can there be a set $S$ of $2 n$ vectors in $V$ such that any $n$ vectors in $S$ span a space of dimension exactly $n-1$, but no $n$ vectors $...
- 20.2k
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Nonnegativity of an integral over the unitary group
For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...
- 1,593
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4
answers
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The derivative of the Cholesky factor
Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am ...
- 620
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Centralizer of a Matrix over a Finite Field
This question in stackExchange remained unanswered.
Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)...
- 801
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History of Sylvester's resultant?
Suppose that we have two polynomials that split:
$$\begin{align*}
f(x)=\sum_{k=0}^d a_{d-k}x^k&=\prod_{i=1}^d (x-\lambda_i),\\
g(x)=\sum_{k=0}^e b_{e-k}x^k&=\prod_{j=1}^e (x-\mu_j).\\
\end{...
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Catalan determinants in search of a proof: Part II
This problem involves the Catalan numbers $C_n=\frac1{n+1}\binom{2n}n$.
I can prove the below equality by computing each of the two sides, directly. That means, there is an algebraic proof.
...
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Do singular values dominate eigenvalues?
Suppose $A$ is an $n \times n$ complex matrix with singular values $s_1 \ge s_2 \ge \cdots \ge s_n$ and eigenvalues $(\lambda_i)_{i=1}^{n}$ arranged so that $|\lambda_1| \ge |\lambda_2| \ge \cdots \...
- 1,135
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Quadratic forms over finite fields
I'm reading some very old papers (by Birch et al) on quadratic forms and I don't get the following point:
If $f$ is a quadratic form in $X_1,X_2,\cdots,X_n$ over a
finite field, then one can ...
- 5,163
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votes
1
answer
863
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Pfaffian equals complex determinant?
Let $V$ be a Euclidean vector space and let $V^{\mathbb{C}} = V \oplus V$ be its complexification, with complex structure
$$J = \begin{pmatrix} 0 & -\mathrm{id}\\ \mathrm{id} & 0 \end{pmatrix}....
- 9,853
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Imaginary eigenvalues
Consider the matrix
$$A(\mu) = \begin{pmatrix} 0 & 1& 0 & 0 \\ -1 & -i\mu & 0 & i \\ 0 & 0 & 0 & 1 \\ 0 &i & -1 & i\mu \end{pmatrix}.$$
This matrix is ...
- 124
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Inverse of a small submatrix
Let $A$ be a large matrix (say, $1000 \times 1000$), and let $\mathcal I = \{2,3,5\}$ be a set of row/column indices. Let $(A^{-1})_{\cal I \times I}$ denote the submatrix of $A^{-1}$ that consists of ...
- 241
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Singular value decomposition over finite fields?
What is the definition of a singular value over a finite field $\mathcal{F}$ of a matrix ${\bf A}$ in $\mathcal{F}^{m\times n}$? Is there a geometric intuition in the same manner as with the real case ...
- 113
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Bounds for the difference in the number of ones in $M$ and $M^{-1}$
If $M$ is a full rank $n$ by $n$ binary matrix over $\mathbb{F}_2$, how much larger or smaller can the number of $1$s in $M^{-1}$ be, compared to the number of $1$s in $M$?
Clearly the identity matrix ...
- 3,377
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Groups of matrices that preserve several quadratic forms
Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least ...
- 2,232
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Maximum singular value of a random $\pm 1$ matrix
Define a matrix $\mathbf{A} \in \mathbb{R}^{m \times n}$ such that each element is independently and randomly chosen with probability $\frac 12$ to be either $+1$, or $-1$. Do you know any result in ...
- 199
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Are these abelian groups free?
Suppose we have a countable, torsion-free abelian group $A$ with the property that for each element $a\neq 0$ the set $D_a=\{x\in A|\exists n\in \mathbb{Z}:nx=a\}$ is finite.
Is $A$ already a free ...
- 11.1k
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Axiom of choice and algebraic tensor product
The first part of the question was asked on Math-stackexchange.
Let $V$, and $W$ be vector spaces. By the universal property of the tensor product,
there is a canonical map from $V^*\otimes W^*$ ...
- 1,035
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A square root inequality for symmetric matrices?
In this post all my matrices will be $\mathbb R^{N\times N}$ symmetric positive semi-definite (psd), but I am also interested in the Hermitian case. In particular the square root $A^{\frac 12}$ of a ...
- 5,380
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Three half circles on the plane may not meet nicely
Let $H$ denote the union of the northern hemisphere of the unit circle $S^{1}$ with the interval $[-1,1]$ on the $x$-axis. That is, $H=\{(x,\sqrt{1-x^{2}}):-1\le x\le 1\}\cup\{(x,0):-1\le x\le 1\}$
...
- 2,136
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What is the size of the category of finite dimensional F_q vector spaces?
The size of a finite skeletal category C in the sense of Leinster is defined as follows: Label the objects of C by integers 1,2,...,n and let aij be the number of morphisms from i to j (for i and j ...
- 2,600
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Encyclopedia of properties of nonnegative matrices
I'd like to buy a book that contains more or less all known properties of elementwise nonnegative nonnegative matrices, i.e. matrices $A$ such that $a_{ij} \ge 0$ for all $1 \le i,j \le n$.
Chapter 8 ...
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Determinant and eigenvalues of a specific matrix
This came up in a conversation with an engineer friend of mine.
Let $c>0$ be a constant. Let $A_{ij}$ be an $n$ by $n$ matrix with entries
$$
A_{ij} = e^{-c(i-j)^2}.
$$
Is there a name for this ...
- 5,186
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Kernel of a matrix and the Catalan numbers
Let $B_n$ denote the Boolean lattice of a set with $n \geq 2$ elements and $C_n$ the matrix with entries $c_{i,j}=1$ if $i \leq j$ and $c_{i,j}=0$ else, where $i,j\in B_n$.
Let $M_n:=C_n+C_n^T$ (this ...
- 26.5k
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Decide if a matrix is transposable
A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...
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Sum of commuting semisimple operators
Let $V$ be a finite dimensional vector space over a field $K$. An operator $T:V\to V$ is called semi-simple if every $T$-invariant subspace of $V$ has a $T$-invariant complement(for algebraically ...
- 2,233
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Existence of a pair of matrices in SL(2,Z) satisfying certain constraints on the spectral radius
Some background: my coauthors and I are working on a problem which deals with the exponential growth rates of certain infinite products of matrices. One of the sub-problems which arises in this ...
- 6,206
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714
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A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
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A variant of Cholesky decomposition involving binary matrices
Studying a problem that is not directly related to linear algebra I came across the following problem.
Let $B$ be $n \times n$ symmetric matrix whose entries are non-negative integers. I would like ...
- 3,463
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How to calculate the inverse of the sum of an identity and a Kronecker product efficiently?
I have a matrix $K$ which is the sum of a identity and a Kronecker product of two symmetric matrices as following and I want to calculate the inverse of it $K^{-1}$.
\begin{eqnarray}
K=\mathbf{I}_{mn}+...
- 607
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Strong approximation for principal ideal domains
A well known consequence of the strong approximation theorem for semisimple simply connected algebraic groups over a number field is that certain reduction maps are surjective, for example, the ...
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1
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How do computer algebra packages like Sagemath implement rank of a matrix
I am not sure if this is the right place to ask this question, but I believe there will be people here who do computations on computer algebra packages like Sage in their work.
I have been using ...
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One question on circulant $\pm1$ matrices
Let $n > 13$ be a positive integer. Is there any $n \times n$ circulant $\pm1$ matrix $A$ satisfying the following property
$$AA^T=(n-1)I+J$$
where $I$ is the $n \times n$ identity matrix and $J$ ...
- 696
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Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
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Diagonal plus low-rank decomposition of symmetric matrices
Let $r(n)$ be the smallest integer such that
All symmetric $n\times n$ matrices with non-zero real entries can be written as the sum of a diagonal matrix and a matrix of rank $r(n)$
What is $r(n)$...
- 302
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a Hankel matrix of involution numbers
Let $I_k$ denote the enumeration of involutions among permutations in $\mathfrak{S}_k$. I always enjoy these numbers. Of course, here is yet another cute experimental finding for which I ask validity. ...
- 43.2k
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Best way to find a closest vector in a lattice
Let $v_1,\dotsc,v_n$ be linearly independent vectors in $\mathbb{R}^n$, and let $\Lambda=\bigoplus_{i=1}^n \mathbb{Z}v_i$. The question is, given a vector $w$ in $\mathbb R^n$, find the element $v$ ...
- 351
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Axiom of choice in linear algebra
It is well known that the Axiom of Choice is needed to prove that every vector space has a basis (in fact this statement is equivalent to the AC). But what about the apparently weaker statement that ...
- 2,049
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votes
3
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591
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Non-singular matrix with restricted entries
Given a set $S$ of integers with $1 \not\in S$, let us consider the set $\mathcal{M}$ of all the symmetric matrices $M$, such that:
All the diagonal entries of $M$ are equal to $1$.
All the off-...
- 571
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How to prove this determinant is positive-II?
Question: Given an arbitrary number of real matrices of the form $ A_i=
\biggl(\begin{matrix}
C_i+E_i & B_i \\
B_i^T & D_i-F_i
\end{matrix} \biggr)
$, where $B_i$ is an arbitrary $n\times n$ ...
- 845
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1k
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An algorithm to find non-trivial linear dependencies
This question is inspired by another MO question about special stratifications of equivariant Grassmannians, that turned out to be a problem of computing non-trivial circuits in a vector matroid. To ...
- 56.6k
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1
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410
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An "existence contra partition of unity" statement for integer matrices?
While reading a blog post on partitions of unity at the Secret Blogging Seminar the following question came into my mind.
Let $n$ be a positive integer and let $B_1$ and $B_2$ be $n \times n$ ...
- 2,600
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votes
2
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558
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Classification of algebras of finite global dimension via determinants of certain 0-1-matrices
I restrict to the elementary problem that is equivalent to give a classification when Morita-Nakayama algebras have finite global dimension (see the end of this post for some background).
A Morita-...
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1
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Exact eigenvalues of a specific tridiagonal matrix
I'm studying the following tri-diagonal matrix
$$
X = \begin{pmatrix}
0 & x_0 & 0 & 0 &\cdots & 0 & 0 & 0 \\\
x_0 & 0 & x_1 & 0 &\cdots & 0 & ...
- 161