Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

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Probability for an $n\times n$ matrix to have only real eigenvalues

Let $A$ be an $n\times n$ random matrix where every entry is i.i.d. and uniformly distributed on $[0,1]$. What is the probability that $A$ has only real eigenvalues? The answer cannot be $0$ or $1$, ...
Exodd's user avatar
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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} & \...
Rofl Ukulus's user avatar
34 votes
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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} ...
trion's user avatar
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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 ...
COTO's user avatar
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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 ...
Sidharth Ghoshal's user avatar
26 votes
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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 ...
spin's user avatar
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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 $\displaystyle\prod_{i=1}^N A_{ii}$?
Ahmed Fasih's user avatar
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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&...
entrelac's user avatar
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18 votes
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Is the power of $2$ in the Euclidean norm related to the fact that the algebraic closure of the reals is $2$-dimensional?

Consider any local field $K$, endowed with its topological field structure. We define the function $| \cdot | : K \to \mathbb{R}_{\ge 0}$ as $$|x| = \frac{\mu(xS)}{\mu(S)},$$ where $\mu$ is any Haar ...
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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 ...
Nick Alger's user avatar
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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 ...
Fiktor's user avatar
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17 votes
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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 ...
Ilan Aizelman WS's user avatar
17 votes
1 answer
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$M\cong N$ iff $[M:N]_R$ is a principal fractional ideal

Let $R$ be a Dedekind ring, $K$ its field of fractions, $U$ a finite vector space over $K$, and $M,N$ finitely generated $R$-modules that span $U$, i.e. contain a basis of $U$. For every $\mathfrak p ...
Rodrigo's user avatar
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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 ...
Morad's user avatar
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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 $\...
user1101010's user avatar
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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 $a_{ij},$ ...
Bobson Dugnutt's user avatar
15 votes
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565 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)+...
Ewan Delanoy's user avatar
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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 $T \otimes S$ ...
1LiterTears's user avatar
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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 &...
Keaton's user avatar
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14 votes
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Proving the Product of Unitary Matrices is also Unitary

This is my attempt so far: Suppose I have two unitary matrices, A and B, such that $A^*A=I$ and $B^*B=I$. We want to show that $(AB)^*(AB)=I$. So we have that $$(AB)^*(AB)=B^*A^*AB=B^*IB=I.$$ Does ...
Walter's user avatar
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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) ...
Simplicius's user avatar
14 votes
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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-...
Fernando's user avatar
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13 votes
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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 $T$ ...
user1101010's user avatar
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526 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 ...
fweth's user avatar
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12 votes
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143 views

What axioms are needed to show that every vector space has a "coordinate system"?

It is a well-known theorem that the axiom of choice (AC) is equivalent to the statement that every vector space over any field has a basis. In this question I am interested in the following somewhat ...
Tim Seifert's user avatar
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A mistake in Bourbaki's construction of general tensor product?

Fortunately someone seems to have a asked a similar question before here: Bourbaki's construction of generalized tensor product of modules ; to save space and time, I will use the same notation ...
user831160's user avatar
12 votes
0 answers
278 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 $...
ryanriess's user avatar
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368 views

A direct proof of the Vandermonde decomposition of a nonsingular Hankel matrix?

I have been doggedly searching for a direct proof of the following theorem: Theorem 1: Let $H$ be a complex nonsingular $n\times n$ Hankel matrix. Then $H$ can be factorized $H = V^\top DV$ where $V$ ...
eepperly16's user avatar
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11 votes
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Clubs whose intersections are multiples of six (Oddtown variant)

This is a question about generalizing the famous "Clubs in Oddtown" problem. The original setup is that a town has $n$ people, and $m$ clubs each consisting of a subset of the population. ...
Mike Earnest's user avatar
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11 votes
0 answers
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Why can't we define traces and determinants for non-square matrices?

I'm reading a book that claims that determinants and traces are only defined for square matrices, but doesn’t really explain why. From a calculation standpoint, this seems correct because I wouldn’t ...
componentvector's user avatar
11 votes
0 answers
95 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 ...
Frank Noam's user avatar
11 votes
0 answers
415 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\...
Hans's user avatar
  • 3,539
11 votes
1 answer
529 views

Determinant of $n\times n$ matrix with parameter

Problem: Let $\delta \in \mathbb{R}^+$ and $n\in \mathbb{N}$. The matrix $A_n = (a_{i,j}) \in \mathbb{R}^{n\times n}$ is defined as $$ a_{i,j} = \prod_{k=0}^{i-2}\left((j-1)\delta +n-k\right) $...
Sil's user avatar
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11 votes
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955 views

Does the Hadamard product of two vectors have a geometric interpretation?

Pointwise multiplication (or element-wise) is also known as the Hadamard product. Is there a geometric interpretation similar to addition or subtraction in vector spaces?
user avatar
11 votes
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547 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 ...
user178543's user avatar
11 votes
0 answers
2k views

Proof of rank-nullity via the first isomorphism theorem

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked? Lemma. If $V$ is a finite-dimensional $F$-vector space ...
user168365's user avatar
11 votes
1 answer
3k views

What does the degree of a matrix minimal polynomial encode?

Let $\mathsf{F}$ be any field. Let $A$ be an $n \times n$ matrix over $\mathsf{F},$ whose rank is $r \le n.$ Let $\mu \in \mathsf{F}[x]$ be the minimal polynomial of $A.$ What does $\deg(\mu)$ tell ...
user avatar
10 votes
0 answers
331 views

Matrix powers of product of diagonalizable and orthogonal matrix

Suppose I have the following matrix constructed from some orthogonal matrix $O$ and a $\pm 1$ diagonal matrix $D=diag(\pm1,\dots,\pm1)$ $$ A = O D O^{-1} D. $$ Is there a simple way to evaluate $A^n$ ...
User71942's user avatar
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10 votes
0 answers
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Linear independence of tensor product basis $\{ v_i \otimes w_j\}$ for $\{v_i\}$ and $\{w_j\}$ linearly independent.

Show that the set $\{v_i \otimes w_j\}$ is a linear independent subset of $V\otimes W$ when $\{v_i\}$ and $\{w_j\}$ are independent subsets of V and W respectively. I want to find an error in a proof ...
mz71's user avatar
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10 votes
0 answers
138 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 ...
Maxime Ramzi's user avatar
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10 votes
0 answers
699 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 ...
Gardenia625's user avatar
10 votes
0 answers
352 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 &...
Graz's user avatar
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10 votes
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438 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 ...
Ludwig's user avatar
  • 2,269
10 votes
3 answers
629 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 ...
Alex Shtoff's user avatar
10 votes
0 answers
196 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}(...
Asaf Shachar's user avatar
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10 votes
0 answers
141 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 ...
user avatar
10 votes
0 answers
195 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 &...
Yly's user avatar
  • 15.3k
10 votes
0 answers
254 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} = ...
user326210's user avatar
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10 votes
1 answer
1k 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 ...
Christian's user avatar
  • 463
10 votes
1 answer
877 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 ...
goblin GONE's user avatar
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