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Questions tagged [linear-algebra]

Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically concerning matrices, use the (matrices) tag. For questions specifically concerning matrix equations, use the (matrix-equations) tag.

180
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10k views

Limit of sequence of growing matrices

Let $$ H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right), $$ $K_1=\left(\...
36
votes
0answers
743 views

What does the space of non-diagonalizable matrices look like?

Let $k$ be a field $\mathbb C$. Consider the action of $G=GL_n(k)$ by conjugation on the set of $n\times n$ matrices over $k$. The collection $X$ of matrices with repeated eigenvalues over $\...
22
votes
0answers
441 views

Determinant of a matrix that contains the first $n^2$ primes.

Let $n$ be an integer and $p_1,\ldots,p_{n^2}$ be the first prime numbers. Writing them down in a matrix $$ \left(\begin{matrix} p_1 & p_2 & \cdots & p_n \\ p_{n+1} & p_{n+2} & \...
20
votes
0answers
1k views

How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
20
votes
0answers
747 views

Determining the kernel of a Vandermonde-like matrix

The kernel of a Vandermonde matrix can be determined using this formula. The following type of matrix has a similar structure, and should also have a one-dimensional kernel. $$V= \begin{bmatrix} ...
18
votes
0answers
460 views

Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
16
votes
0answers
391 views

Upper triangular matrices $B$ that commute with every upper triangular matrix commuting with $A$

I remember being told that this was true by a professor, but I haven't been able to find a source for it yet. In the theorem as stated, $\mathbb{F}$ is any field and $T_n(\mathbb{F})$ denotes the ...
16
votes
0answers
313 views

Can some proof that $\det(A) \ne 0$ be checked faster than matrix multiplication?

We can compute a determinant of an $n \times n$ matrix in $O(n^3)$ operations in several ways, for example by LU decomposition. It's also known (see, e.g., Wikipedia) that if we can multiply two $n \...
16
votes
0answers
456 views

The longest list of analogies between vector spaces and categories ever made

I suspect this question exists in different forms, elsewhere. I would like to know what's going on with this table, how to fill the missing items and how to continue the list, and what is the ...
15
votes
0answers
743 views

Conjecture about normal matrices with magnitude symmetry

Given a complex, normal matrix A such that $|A_{ij}| = |A_{ji}|$ for all i,j I conjecture that A must either be of the form: 1) $e^{i\theta}H$ where H is a hermitian matrix and $\theta$ is real. (...
15
votes
0answers
879 views

Matrix generated by prime numbers

Let $p$ be the vector of dimension $n^2$ consisting of ordered prime numbers i.e. $p= [ 1 \ 2 \ 3 \ 5 \ 7 \ldots]^T$ and $A$ be the matrix of dimension $n\times{n}$ constructed with this vector by ...
14
votes
0answers
215 views

Find the cardinality of a subset of $GL_n( \bf F_p)$

Let $m,n \in \bf N$.Let $\bf F_p$ denote the prime field of characteristic $p$.Consider the set $$ X_m = \{A \in GL_n( {\bf F_p}): A^m=1 \}$$ Compute the cardinality of $X_m$. Its clear that $\vert ...
12
votes
0answers
229 views

Fastest way to check existence of solution for a linear system of inequalities

What is the fastest way to check if there exists a solution to the inequality $A x \leq b$, with $A \in \mathbb R^{n \times m}$? I know this can be checked through the phase 1 of a linear programming ...
12
votes
0answers
6k views

Simpson's Rule for Double Integrals

Simpson's Rule for double integrals: $$\int_a^b\int_c^df(x,y) \,dx \,dy$$ is given by $$S_{mn}=\frac{(b-a)(d-c)}{9mn} \sum_{i,j=0,0}^{m,n} W_{i+1,j+1} f(x_i,y_j) $$ where: $$W= \begin{pmatrix} 1&...
12
votes
0answers
285 views

Linear functional equation

During my mathematical musings I encountered the following functional equation : denote by $L$ the set of all functions ${\mathbb Z}^2 \to {\mathbb C}$ satisfying $$ \begin{array}{cl} &f(x+a,y+b)+...
12
votes
0answers
447 views

Matrix diagonalization theorems and counterexamples: reference-request.

I'm looking for exhaustive list of diagonalization theorems and counterexamples in linear algebra. In this question I understand the question of matrix diagonalization very broadly: suppose we have ...
12
votes
0answers
4k views

Tensor Product is associative, distributive, not commutative.

Tensor Product is associative, distributive, not commutative. Here is my attempt to show tensor product is associative, is it legit? If $T$ is a $p$-tensor and $S$ a $q$ tensor, then $T \otimes S$ ...
12
votes
0answers
957 views

Inverse of Toeplitz Matrix Property

Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form $$\left[\begin{array}{llll} a_0 & a_1 & \dots & a_n\\ a_1 & a_0 &...
12
votes
0answers
815 views

Pfaffian properties

Given a $2n\times 2n$ real skew-symmetric matrix $A$, its Pfaffian $\mathrm{Pf}$ is defined as: $$ \mathrm{Pf}(A) = \frac1{2^n n!}\sum_{\sigma\in S_{2n}}\mathrm{sgn}(\sigma)\prod_{i=1}^n A_{\sigma(2i-...
11
votes
0answers
226 views

Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also have the condition: for any collection of $\...
11
votes
0answers
296 views

A conjecture on the Lyapunov equation

Let $A\in\mathbb{R}^{n\times n}$ be a Hurwitz stable matrix (i.e., all the eigenvalues of $A$ have strictly negative real part). Let $X\in\mathbb{R}^{n\times n}$ be a positive semi-definite matrix of ...
11
votes
0answers
110 views

Coordinate free proof that trace$(A)=0$ $\Rightarrow$ $A=BC-CB$.

As you probaby know, the trace function on square matrices has the property that $\text{trace}(AB-BA)=0$. You might also know that the converse is true, $\text{trace}(A)=0$ imples $A=BC-CB$ for some ...
11
votes
0answers
108 views

Tensor Product and Physics

During lecture, my abstract algebra professor was saying that the exactness of the tensor product is "absolutely essential" to the existence of physical phenomena such as black holes and the big bang. ...
11
votes
0answers
207 views

Finding the ratio between two $8$-dimensional volumes

EDIT: At this point, geometric interpretations of conditions 2-4 would qualify as an answer. This can include symmetries of the region. I have a real $3 \times 3$ matrix $A$ with entries $a_{ij},$ ...
11
votes
0answers
342 views

given the inverse of a matrix, is there an efficient way to find the determinant?

Suppose one has the inverse $A^{-1}$ of an $N\times N$ non-singular matrix $A$. Is there an ''efficient'' way to obtain $\det{A}$? With ''efficient'' I mean anything that has a better scaling ...
11
votes
0answers
152 views

$\text{SL}(2, \mathbb{F}_q)$, for which characters is the $G$-representation irreducible?

Followup to here. Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes ...
11
votes
0answers
514 views

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
10
votes
0answers
103 views

Does the space of matrices above rank $k$ admit a transitive Lie group action?

$\newcommand{\End}{\operatorname{End}}$ $\newcommand{\GL}{\operatorname{GL}}$ Let $V$ be a real $d$-dimensional vector space ($d \ge 4$). Let $2 \le k \le d-2$. Define $H_{>k}=\{ A \in \text{End}(...
10
votes
0answers
192 views

Is there a sense in which the Chi-squared distribution is an inner product?

I have been self-studying statistics recently, and the apparent similarities between linear algebra (especially Hilbert spaces) and statistics have been popping out to me. Linear independence gets ...
10
votes
0answers
257 views

Bounding the minimum singular value of a block triangular matrix

Question: What is the sharpest known lower bound for the minimum singular value of the block triangular matrix $$M:=\begin{bmatrix} A & B \\ 0 & D \end{bmatrix}$$ in terms of the properties ...
10
votes
0answers
355 views

Flipping Summation of Kronecker Products

Question Suppose $\mathbf A$ is an $n\times n$ matrix, and that $\mathbf B_i$ is an $m\times m$ matrix, for all $i\in\{1,\dots, n\}$. Is it possible to find $n\times n$ matrices $\mathbf U$ and $\...
10
votes
0answers
487 views

Lowest dimensional faithful representation of a finite group

How does one compute the lowest dimensional faithful representation of a finite group? This question originated in the context of given a finite group G: trying to find the lowest dimensional shape ...
10
votes
0answers
209 views

How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
10
votes
0answers
200 views

Counting the number of elements in a double coset

Let $G$ denote the groups of $n\times n$ invertible matrices and $H$ be the subgroup of invertible upper triangular matrices. For $n=2$, by row reduction, or equivalently LU decomposition, it is ...
10
votes
0answers
2k views

Simultaneous diagonalization of quadratic forms

I would like to collect references (or direct quotations) about as many "simultaneous diagonalization" results in linear algebra as possible. Let $V$ be an $n$-dimentional ($n$ finite) vector space ...
9
votes
0answers
147 views

Reference request: arranging distinct numbers into a full rank matrix

I thought of the following problem: Let $n\ge 2$. Suppose you have $n^2$ distinct numbers in some field. Is it necessarily possible to arrange the numbers into an $n\times n$ matrix of full ...
9
votes
0answers
99 views

What is $\ \overline{\bigcup_{p≥ 1}\ \{A\in M_n(\mathbb C), \ A^p = I_n\}} \ $?

Let $\Gamma_p = \{A\in M_n(\mathbb C), A^p = I_n\}$ and let $\Gamma = \bigcup_{p≥ 1}\ \Gamma_p$. What is the closure of $\Gamma$ ? (This is from an oral exam). Let $B \in M_n(\mathbb C)$ such that ...
9
votes
0answers
257 views

Number of simultaneously solvable linear equations with nonnegative variables

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$. There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...
9
votes
0answers
186 views

Convergence of a linear recurrence equation

Let $T \colon \mathbb{C}^n \to \mathbb{C}^n$ be a linear operator. Let $\{u_k\} \subset \mathbb{C}^n$ and $\{v_k\} \subset \mathbb{C}^n$ be two sequences of vectors. Suppose the spectral radius of $T$ ...
9
votes
0answers
142 views

Matrix exponential, containing a thermal state

Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 & \cdots \\ 0 & 2 & 0 & -3 & \cdots \\ 0 & 0 & 3 &...
9
votes
0answers
107 views

Finite group of “linear substitutions”

From what I can tell, a linear substitution is an operation on a set of variables $x_1,\ldots,x_n$ which sends them to a new set of variables $y_1,\ldots, y_n$ via a linear transformation $$\vec{y} = ...
9
votes
0answers
165 views

A wrong proof for an (evident) lemma

(Eliashberg, Y.; Mishachev, N.M., Wrinkling of smooth mappings and its applications. I, Invent. Math. 130, No.2, 345-369 (1997). ZBL0896.58010. \cite{EM}) Let $ \alpha : [a, b] \to \mathbb{R}$ is ...
9
votes
0answers
349 views

Prove that the group $\mathrm{GL}(n, \mathbb{Z})$ is finitely generated

Knowing that for $n \geq 2$, $\mathrm{GL}(n, \mathbb{Z}) = \big\{ A \in \mathrm{M}_{n,n}(\mathbb{Z}) \mid \det(A) \in \{ 1, −1 \} \big\}$ is a group with respect to matrix multiplication, prove that ...
9
votes
0answers
394 views

Theoretical link between the graph diffusion/heat kernel and spectral clustering

The graph diffusion kernel of a graph is the exponential of its Laplacian $\exp(-\beta L)$ (or a similar expression depending on how you define the kernel). If you have labels on some vertices, you ...
8
votes
0answers
260 views

Optimal orthonormal basis to represent a Gaussian

I am looking at representing a set of Gaussians, of the form $\exp(-\frac{(r-r_i)^2}{2 \sigma^2})$, on a 1D domain. I do not know $r_i$ and $\sigma$ prior to defining the basis $\{ \phi_k(r) \}_{k=1}^...
8
votes
0answers
234 views

When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is related to the question How many connected components for the intersection $S \cap GL_n(\mathbb R)$ where $S \subset M_n(\mathbb R)$ is a linear subspace? I asked. There is a nice example in ...
8
votes
0answers
87 views

A "geometric''-ish infinite sum of matrices

Suppose I have full rank $n\times n$ matrix $A$ with $\rho(A) < 1$ and I want to find an expression for $$S = X + A^\top X A + A^{2\top} X A^2 + A^{3\top} X A^3 + \dots$$ where $X$ is an $n\...
8
votes
0answers
154 views

Bases in vector spaces without $AC$

It is known that without the axiom of choice, not every vector space has a basis. But I was wondering, if I don't assume the axiom of choice, and I choose a vector space $V$ which does have a basis (...
8
votes
0answers
198 views

Questions on color theory, expressed in linear algebra

I'm reading into color theory and there were a few questions which I asked myself along the way, maybe you can put me forward to some source where I can find answers or give them directly. The ...
8
votes
0answers
4k views

Matrix multiplication of columns times rows instead of rows times columns

In ordinary matrix multiplication $AB$ where we multiply each column $b_{i}$ by $A$, each resulting column of $AB$ can be viewed as a linear combination of $A$. If however if we decided to multiply ...