Questions tagged [linear-algebra]

For questions about 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.

24,710 questions with no upvoted or accepted answers
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294
votes
0answers
16k 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(\...
34
votes
0answers
972 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} & \...
30
votes
0answers
436 views

probability for a $n\times n$ matrix to have no complex eigenvalues

Let $A$ be a $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has no complex eigenvalues? The answer cannot be 0 or 1, since ...
29
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0answers
1k 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} ...
27
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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 ...
22
votes
0answers
426 views

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

Let $m,n \in \mathbb N$. Let $\mathbb F_p$ denote the prime field of characteristic $p$. Consider the set $$ X_m = \{A \in GL_n( {\mathbb F_p}): A^m=1 \}$$ Compute the cardinality of $X_m$. Its ...
20
votes
0answers
711 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 ...
19
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0answers
2k 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 ...
18
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0answers
849 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 ...
16
votes
0answers
380 views

Coordinate free proof that $\operatorname{trace}(A)=0\:\Longrightarrow\:A=BC-CB$

As you probably know, the trace function on square matrices has the property that $$\operatorname{trace}(AB-BA)=0\,.$$ You might also know that the converse is true: $$\operatorname{trace}(A)=0\;\text{...
16
votes
0answers
8k 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&...
14
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0answers
313 views

Toeplitz matrices question with Fourier coefficients

Denote: $f(e^{i\theta})$ is continuous and strictly positive on the interval $ 0 \le \theta \le 2\pi$ with Fourier coefficients $$ t_j = \frac{1}{2\pi}\int_0^{2\pi}f(e^{i\theta})e^{-ij\theta} \quad ...
14
votes
0answers
541 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 ...
14
votes
0answers
499 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)+...
14
votes
0answers
569 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 ...
14
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0answers
5k 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 $T \otimes S$ ...
13
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0answers
250 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 $a_{ij},$ ...
13
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0answers
1k 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
251 views

A puzzling KKT for LMI vs. scalar constraint

I am trying to understand the KKT conditions for LMI constraints in order to solve my original question in KKT conditions for $\max \log \det(X)$ with LMI constraints. In the meantime, I found a much ...
12
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0answers
250 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 $\...
12
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0answers
166 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 ...
12
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0answers
8k views

What do you call the product of a matrix's diagonal elements?

The trace of $A$, an $N\times N$ matrix, is $\displaystyle\sum_{i=1}^N A_{ii}$. What do you call $\displaystyle\prod_{i=1}^N A_{ii}$?
12
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0answers
3k 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) ...
12
votes
0answers
1k 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
1answer
233 views

Size of a linear image of a cube in $\mathbb{Z}^d$

Suppose that we have an element $v = (v_1, \dots, v_d) \in \mathbb{Z}^d$ such that $\gcd(v_1, \dots, v_d) = 1$. Then $v$ is contained in some base of $\mathbb{Z}^d$ (seen as a free-abelian group or a ...
11
votes
0answers
299 views

Studying representations of orthogonal group via symmetric group?

Let $V$ be the vector space of $n\times n$ symmetric matrices with real entries. On $V$ we have the natural action of the orthogonal group $\textrm{O}(n)$ defined by $g.A:=g\cdot A\cdot g^{t}$ for $g\...
11
votes
0answers
285 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 $T$ ...
11
votes
0answers
353 views

How do you solve linear least-squares modulo $2 \pi$?

I have an overdetermined system of $m$ equations ($i = 1, 2, \dots, m$) $$ \sum_{j=1}^n A_{ij} \, x_j = y_i \pmod{2\pi} $$ where the $x$ coefficients are unknown, and $m > n$. This is, essentially, ...
11
votes
0answers
369 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 ...
11
votes
0answers
530 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
231 views

Can an orthogonal matrix move monotonically toward a signed permutation matrix?

The question is motivated by this question on Mathematics SE. Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
10
votes
0answers
70 views

Identities for subspaces and linear maps

Wikipedia has a nice list of identities for how intersections, unions, and complements interact with images and preimages of set functions. But if $f:V \to W$ is a linear map, many of the identities ...
10
votes
0answers
104 views

Exemples of applications of “groupoidification” to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
10
votes
0answers
378 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 ...
10
votes
3answers
482 views

Box-constrained orthogonal matrix

Given constants $\ell, u \in \mathbb{R}^{3 \times 3}$ and the following system of constraints in $P \in \mathbb{R}^{3 \times 3}$ $$ P^T P = I_{3 \times 3},\quad \ell_{ij} \leq P_{ij} \leq u_{ij}, $$ I ...
10
votes
0answers
144 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
162 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 &...
10
votes
0answers
158 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} = ...
10
votes
1answer
615 views

Generalisation of prime numbers to matrices?

Is it possible to generalise prime numbers to matrices? I'm trying to solve a Rubix cube in the minimum number of steps and I think this would be useful. I think it's possible to represent Rubix cube ...
10
votes
1answer
500 views

Is there a name for the group of real matrices whose determinant is an element of $\pm 1$?

The group of matrices whose determinant is non-zero is called the "general linear group", and the group of matrices whose determinant is $1$ is called the "special linear group". In between these two ...
10
votes
0answers
420 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
1answer
137 views

In a finite dimensional inner product space with $T ∈ L(V)$, show that $\langle u,v\rangle = \langle T(u),T(v)\rangle$ implies $T$ is invertible.

Here is how I've tried to go about it, and I'm curious if it's true or if I'm way off base. T is invertible iff null$(T)=\{0\}$. Let $v∈V$ and suppose $T(v)=0$. If we can show that $v=0$, then $T$ is ...
9
votes
0answers
421 views

If nilpotent matrix $A$ and $AB−BA$ commute, show that $AB$ is nilpotent.

Let $A$ and $B$ be $n×n$ complex matrices. If $A$ is a nilpotent matrix, and $A$ commute with $AB−BA$, show that $AB$ is nilpotent. Equivalently, the question can be expressed as following ...
9
votes
0answers
217 views

Determinant of a symmetric matrix with entries on diagonals

I am interested in the calculation of the determinant of the $N\times N$ symmetric matrix \begin{equation*} \mathbf B = \left(\begin{array}{*{20}c} 2 & & -1& &-1& &\\ & 2 &...
9
votes
0answers
77 views

Matrix performing local differintegral analysis being its own inverse. Coincidence?

I found a curious matrix $$T = \begin{bmatrix}1&2&1\\1&0&-1\\1&-2&1\end{bmatrix}$$ This matrix (or actually $\frac 1 2 T$) performs Local mean value (integral) estimation. ...
9
votes
1answer
151 views

How many connected components could the intersection of $\{A \in M_n(\mathbb R): \rho(A) < 1\}$ and an affine subspace in $M_n(\mathbb R)$ have?

Let $\mathcal E = \{A \in M_n(\mathbb R): \rho(A) < 1\}$ where $\rho(\cdot)$ is the spectral radius and $\mathcal U$ be an affine space in $M_n(\mathbb R)$. If we assume $\mathcal E \cap \mathcal U ...
9
votes
0answers
153 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
1answer
242 views

Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

The set of Schur stable matrices is \begin{align*} \mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\}, \end{align*} where $\rho(\cdot)$ denotes the spectral radius of a matrix and the set of ...
9
votes
0answers
119 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
182 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 (...

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