Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
586
questions
4
votes
2
answers
841
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Partitioning an orthogonal matrix into full rank square submatrices
Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of
$U$, does there always exist a corresponding partition ${\mathcal ...
- 81
3
votes
2
answers
670
views
An inequality for the spectral radius of block matrices
Let $d,m$ be positive integers. Suppose that $A_{i,j}$ is a $d\times d$-matrix with real entries whenever $i,j\in\{1,\dots m\}$.
Let $A$ be the $dm\times dm$ matrix that can be written as a block ...
- 27.8k
3
votes
2
answers
406
views
Maximum eigenvalue of a covariance matrix of Brownian motion
$$ A := \begin{pmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{2} & \frac{1}{2} & \frac{1}{3} & \cdots & \frac{1}{n}\\
\frac{1}{3} & \frac{...
1
vote
1
answer
182
views
Factorizing the doubly stochastic matrix where all entries are equal such that the factors are all convex combinations of few permutation matrices
Let $N_{n}=(1/n)_{i=1,j=1}^{n}$ be the $n\times n$-matrix where all the entries are equal.
Suppose $n>0$. Let $\delta_{n}$ be the least natural number such that $N_{n}$ can be factored as $N_{n}=A_{...
- 27.8k
103
votes
15
answers
12k
views
Why are matrices ubiquitous but hypermatrices rare?
I am puzzled by the amazing utility and therefore ubiquity of
two-dimensional matrices in comparison to the relative
paucity of multidimensional arrays of numbers, hypermatrices.
Of course ...
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62
votes
25
answers
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Linear Algebra Texts?
Can anyone suggest a relatively gentle linear algebra text that integrates vector spaces and matrix algebra right from the start? I've found in the past that students react in very negative ways to ...
62
votes
9
answers
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Can a vector space over an infinite field be a finite union of proper subspaces?
Can a (possibly infinite-dimensional) vector space ever be a finite union of proper subspaces?
If the ground field is finite, then any finite-dimensional vector space is finite as a set, so there are ...
- 23.7k
55
votes
21
answers
14k
views
Linear algebra proofs in combinatorics?
Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Some examples of combinatorial theorems with linear algebra proofs are the (weak) perfect graph theorem, ...
52
votes
9
answers
12k
views
Is there a preferable convention for defining the wedge product?
There are different conventions for defininig the wedge product $\wedge$.
In Kobayashi-Nomizu, there is $\alpha\wedge\beta:=Alt(\alpha\otimes\beta)$,
in Spivak, we find $\alpha\wedge\beta:=\frac{(k+l)...
- 4,266
51
votes
2
answers
4k
views
A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
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votes
3
answers
2k
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Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,743
36
votes
5
answers
10k
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Is there a version of inclusion/exclusion for vector spaces?
I am asking for a way to compute the rank of the 'join' of a bunch of subspaces whose pairwise intersections might be non-zero. So in the case n=2 this is just $\dim(A_1+A_2) = \dim(A_1) + \dim(A_2) - ...
- 509
35
votes
2
answers
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Eigenvalues of the product of two symmetric matrices
This is mostly a reference request, as this must be well-known!
Let $A$ and $B$ be two real symmetric matrices, one of which is positive definite. Then it is easy to see that the product $AB$ (or $BA$...
- 2,542
35
votes
2
answers
1k
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Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 151k
34
votes
8
answers
4k
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Uncountable counterexamples in algebra
In functional analysis, there are many examples of things that "go wrong" in the nonseparable setting. For instance, my favorite version of the spectral theorem only works for operators on a ...
32
votes
5
answers
15k
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Is a vector space naturally isomorphic to its dual? [closed]
This question may not be as easy to answer as you think! Some tangentially-related questions have appeared on math.stackexchange but I'm not really convinced by the answers.
In the sequel I will ...
- 2,775
30
votes
2
answers
2k
views
When is $SL(n,R) \rightarrow SL(n,R/q)$ surjective?
Let $R$ be a commutative ring with unit and let $q$ be an ideal of $R$. There is thus a natural map $SL(n,R) \rightarrow SL(n,R/q)$ for all $n$. This map is surjective if $SL(n,R/q)$ is generated by ...
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votes
1
answer
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If $V$ is a vector space with a basis. $W\subseteq V$ has to have a basis too?
Suppose $V$ is a vector space, we say that $\mathcal B$ is a basis for $V$ if:
Every $v\in V$ can be written as a linear combination of elements of $\mathcal B$;
If $\sum\alpha_i b_i = 0$, where $\...
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26
votes
5
answers
1k
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Is the matrix $\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}$ nonsingular?
Suppose we have a $(2m-1) \times (2m-1)$ matrix defined as follows:
$$\left({2m\choose 2j-i}\right)_{i,j=1}^{2m-1}.$$
For example, if $m=3$, the matrix is
$$\begin{pmatrix}6 & 20 & 6& 0 ...
- 1,091
23
votes
5
answers
3k
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Smallest non-zero eigenvalue of a (0,1) matrix
What's the smallest absolute value possible of a non-zero eigenvalue of an $n$ by $n$ square matrix whose entries are either $0$ or $1$ (all operations are over $\mathbb{R}$)? I would be interested ...
- 3,195
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votes
8
answers
3k
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Finitely presented sub-groups of $\operatorname{GL}(n,C)$
Here are two questions about finitely generated and finitely presented groups (FP):
Is there an example of an FP group that does not admit a homomorphism to $\operatorname{GL}(n,C)$ with trivial ...
- 28.8k
20
votes
3
answers
1k
views
Simultaneous "orthonormalization" in $\mathbb{C}^4$
Let $A$ be a positive, invertible $4 \times 4$ hermitian complex matrix.
So we have a positive sesquilinear form $\langle Av,w\rangle$. Say that a pair $(v,w)$ of vectors in $\mathbb{C}^4$ is good ...
- 42.2k
17
votes
3
answers
864
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 1,990
16
votes
4
answers
3k
views
How many minors I need to check to conclude all minors will vanish ?
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish.
I remember hearing that one really does not need to check all possible minors in order to conclude ...
- 1,775
16
votes
1
answer
842
views
Hankel determinants of binomial coefficients
For $\{h_{n}\}_{n=0}^{\infty}$ a real sequence, denote by $H_{n}$ the $n\times n$ Hankel matrix of the form
$$
H_{n}:=\begin{pmatrix}
h_{0} & h_{1} & \dots & h_{n-1}\\
h_{1} & ...
- 2,188
15
votes
3
answers
17k
views
angle between subspaces
Let $E$ be a finite dimensional real inner product space. I want to define the angle between two subspaces $E_1$ and $E_2$. This has a fairly obvious meaning if $E_1$ is 1-diemsnional: Take the ...
- 421
15
votes
0
answers
443
views
The rank of a "triangle-free" matrix
This is a version of the question I asked recently, but the assumptions got now strengthened substantially.
Suppose that $A=(a_{ij})_{1\le i,j\le n}$ is a square matrix with all elements in $\{0,\...
- 22.8k
15
votes
3
answers
23k
views
How to solve this quadratic matrix equation?
I would like to solve for $X$ in the matrix equation
$$
XCX + AX = I
$$
where all the matrices are $n\times n$, have real components, $X$ is positive semidefinite and $C$ is symmetric. My (possibly ...
- 385
14
votes
4
answers
2k
views
Eigenvectors of a particular transition matrix
I am considering a Markov chain with $n$ states with a particularly nice structure. The transition matrix is as follows:
\begin{equation}\mathbf{P}=\begin{pmatrix}
0 & 0& \dots&0 & 0 &...
- 41
14
votes
2
answers
877
views
Involutions in GL_n(Z)
Is there a classification of involutions in $\text{GL}_n(\mathbb{Z})$?
Here's some more details about what I mean. Consider $f \in \text{GL}_n(\mathbb{Z})$ such that $f^2=1$. Regard $f$ as an ...
- 245
13
votes
2
answers
691
views
in search of a transformation between determinants
Motivated by this MO question. Consider the two matrices $A_n$ and $B_n$ with entries $\binom{2j}i$ and $\binom{n+1}{2j-i}$, respectively; for $1\leq i, \,j\leq n$.
I can show $\det A_n=\det B_n=2^{\...
- 41.8k
13
votes
1
answer
1k
views
Is the theory of vector bundles just linear algebra done in a suitable topos?
Sheaves of sets on a space are somehow "parametrized sets". This is the philosophy by which one can do mathematics internal to a sheaf topos (of which theory I admit I know essentially nothing), with ...
- 22.7k
13
votes
0
answers
998
views
Pointwise (Hadamard) matrix product and the rank
$\DeclareMathOperator{\rk}{rk}$
Suppose that $A$ is a square matrix of order $n$. If, for any polynomials $P$ and $Q$ with $\deg P+\deg Q\le 2$, we have
$$ P(A)\circ Q(A^t) = P(1)Q(1)\, I_n \tag{$\...
- 22.8k
12
votes
7
answers
4k
views
Status of the Hadamard Circulant conjecture
The following feels like a community wiki question, so I do it here:
Recently we have heard of a new proof of the Circulant Hadamard conjecture of Ryser
(a long standing difficult conjecture):
...
12
votes
5
answers
1k
views
Does k(X) have a k-basis for every set X, without AC?
This question is inspired by Pace Nielsen's recent question Does a left basis imply a right basis, without AC?.
For any field $k$, the field $k(x)$ of rational functions in one variable has an ...
- 33.9k
11
votes
2
answers
703
views
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 $A_n:=I_n-M_n^2$. ...
- 41.8k
11
votes
4
answers
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views
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
11
votes
1
answer
330
views
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. ...
- 41.8k
11
votes
2
answers
551
views
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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10
votes
1
answer
618
views
$2 \times 2$ 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 ...
- 42.2k
10
votes
2
answers
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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
10
votes
1
answer
912
views
Conjugation between commutative subalgebras of a matrix algebra?
Let $K$ be an algebraically closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two isomorphic commutative subalgebras of $M_n(K)$, it is true that there ...
- 545
10
votes
4
answers
4k
views
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) - $\sum_{i=0}^{n}{n \choose i}_{q}=?$
- 119
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votes
3
answers
482
views
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 \...
- 19.4k
10
votes
3
answers
816
views
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_{\...
- 1,583
10
votes
1
answer
790
views
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
9
votes
3
answers
350
views
Spectrum of orthogonality graph (2)
The orthogonality graph, $\Omega(n)$, has vertex set the set of $\pm 1$ vectors of length $n$, with orthogonal vectors being adjacent.
I am only interested when $4|n$, since otherwise $\Omega(n)$ is ...
- 421
8
votes
1
answer
2k
views
Symplectic block-diagonalization of a real symmetric Hamiltonian matrix
Given a $2n\times 2n$ real, symmetric, Hamiltonian matrix $W$ (anticommutes with the symplectic metric), is there an orthogonal, symplectic matrix $R$ such that $R^\top WR$ is block-diagonal?
Being ...
- 315
8
votes
1
answer
470
views
Coordinate free isomorphism between $d+1$-dimensional antisymmetric rank $2$ tensors and $d$-dimensional symmetric rank $2$ tensors
The space of $(d+1)$-dimensional antisymmetric matrices has the same dimension as the space of $d$-dimensional symmetric matrices, $\frac12d(d+1)$. There are isomorphisms between the two spaces, e.g. ...
- 245
8
votes
1
answer
318
views
"Almost Hankelized" numerical Vandermonde
One of the more utilized determinant is that of Vandermonde's
$$\begin{vmatrix}
1&x_1&x_1^2&\dots&x_1^{n-1}\\
1&x_2&x_2^2&\dots&x_2^{n-1}\\
\ldots&\ldots&\...
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