Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
6,104 questions
7
votes
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answer
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views
A basis-free formula for the determinant as a polynomial
Let $V$ be an $n$-dimensional $\mathbb{K}$-vector space. By a simple calculus trick (*) on homogeneous functions of degree $n$ the determinant is a linear map on the $n$-th symmetric power of the ...
- 1,248
4
votes
1
answer
208
views
Non-increasing property of a norm-like function over matrices
Let $P,Q$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
8
votes
1
answer
428
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Conjecture about the decompositions of the powers of a matrix
Let $U=\{A\in M_3(\mathbb{C});I_3,A,A^2 \;\text{are linearly independent}\}$;
if $3\leq p<q<r$, then let $f(p,q,r)=\{A\in U; A^p,A^q,A^r\in \operatorname{span}(I,A)\}$.
Conjecture. If $A$ is in ...
- 3,941
1
vote
1
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Question on monotonicity of a norm-like function for matrices
Let $P_1,P_2$ be two real orthogonal projections on $\mathbb R^n$, and assume that they are permutation similar. More specifically, assume that each of them is permutation similar to a block diagonal ...
- 857
4
votes
1
answer
173
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Question on power-nonnegative matrix
Let $P_1,P_2\in M_n(\mathbb R)$ be two orthogonal projections, i.e., $P_1^2=P_1=P_1', P_2^2=P_2=P_2'$ and assume that they are unitarily similar.
Let
$$
A=(P_1P_2)\circ(P_2P_1),
$$
where $\circ$ ...
- 857
1
vote
1
answer
60
views
Efficient construction of an orthogonal matrix with prescribed diagonal quadratic forms
Let $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_n)$ and $\mathbf{d}=(d_1,\dots,d_n)$ be two vectors of real numbers sorted in non‑increasing order, satisfying the majorization conditions
$$
\sum_{i=...
- 4,190
2
votes
0
answers
111
views
Finding spectral radius of a polynomial related to a graph
I am trying to find the largest root of the polynomial $$p(x) = (x^2-a)[(x^2-b-1)(x^2-a-1)-(x^2-a)]-(x^2-a+4)(x^2-a-1),$$ where $a,b$ are real constants.
But the problem is the polynomial has degree $...
- 219
3
votes
0
answers
76
views
Eigenvalues and characteristic polynomials of almost-principal minors
Discussion continued and significantly inspired from this MSE question.
Suppose we have computed through the Faddeev-Leverrier algorithm the characteristic polynomial of a matrix $A$: that is, not ...
- 644
4
votes
0
answers
95
views
(Homological) interpretation of the Moore penrose number
Let $A=KQ/I$ be a finite dimensional quiver algebra with connected acyclic quiver $Q$ and admissible relations $I$.
Let $W$ be the Cartan matrix of $A$ (which we can assume to be lower triangular with ...
- 28.5k
1
vote
0
answers
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views
Classification of nilpotent matrices up to orthogonal similarity
Call a matrix $M$ over a field $K$ orthogonal if $M M^T=id$.
Question 1: Is there a classification of nilpotent matrices up to orthogonal similarity (at least for certain fields K), where two ...
- 28.5k
1
vote
0
answers
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views
Nonconvex geometric counting
Fix $d < n$. Let $V \subseteq \mathbb R^n$ be a $d$-dimensional linear subspace. We say that a set $S \subseteq [n]$ is good if there exists a unit-norm $v \in V$ such that $v_i^2 > 1/s$ for all ...
- 49
4
votes
1
answer
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The 'discriminant' of a matrix over a finite field
Let $\mathbb{F}_q$ be a finite field. Let $V=\mathbb{F}_q^m$.
Let $L_1,\dots, L_n$ be the one-dimensional subspaces of $V$, where $n=(q^m-1)/(q-1)$. Let $0\neq e_i\in L_i$ be non-zero representatives, ...
- 2,603
0
votes
0
answers
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views
Explicit orthogonal matrix with prescribed diagonal for the path graph Laplacian
Let $N \ge 2$ and consider the diagonal matrices
$$
\mathbf{D} = \operatorname{diag}(\lambda_0,\lambda_1,\dots,\lambda_{N-1}), \qquad
\mathbf{D}_0 = \operatorname{diag}\!\left(0,\frac{4}{N},\frac{8}{...
- 4,190
3
votes
0
answers
120
views
Proving a uniform bound $\varphi(s)<3$ for a walk-indexed linear program
Fix a balanced walk
$$
s=(s_0,\dots,s_{2n-1})\in\{-1, 1\}^{2n}
$$
with
$$
\sum_{i=0}^{2n-1}s_i=0.
$$
Define the partial sums
$$
x_j=\sum_{i=0}^j s_i\qquad (0\le j\le 2n-2).
$$
Define matrices $T,C\in\...
- 1,012
2
votes
0
answers
65
views
Characterization of spacelike simplices in $1+n$-dimensional Minkowski
Let $\mathbb M^n = \mathbb R^{1,n-1}$ be $n$-dimensional Minkowski space and $\eta\colon \mathbb M^n \times \mathbb M^n \to \mathbb R $ the corresponding indefinite inner product.
How do I see for a ...
- 221