Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,680
questions
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Linear independence of +/- 1 strings/vectors
Let $V=\left\{-1,1\right\}^{n}$. Consider three vectors $v_1,v_2,v_3\in V$. I would like to know whether these vectors are linearly independent over $\mathbb{Z}$. To be more precise - I need a ...
12
votes
5
answers
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Analogue of Cayley Hamilton theorem for operators on Hilbert space
Is there an analogue of Cayley Hamilton theorem which holds for operators on a separable Hilbert space. Obviously the characteristic polynomial will be replaced by something else.
6
votes
1
answer
741
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Dimension of the span of all partial derivatives of a given homogeneous symmetric polynomial $f$ and the polynomial $E(f)$
I need some help about the problem below.
Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set
$$ E(f):=\sum_{j=1}^{...
3
votes
3
answers
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A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
1
vote
1
answer
465
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Projectors onto the invariant subspaces of a unitary representation $U \otimes U^* \otimes U \otimes U^*$
Let $$U \mapsto U \otimes U^* \otimes U \otimes U^*$$ be a unitary representation of the unitary group $U(n)$ acting on the vector space $V$ (where $U^*$ is the complex conjugate of $U$). We can ...
63
votes
7
answers
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How to prove this determinant is positive?
Given matrices
$$A_i= \biggl(\begin{matrix}
0 & B_i \\
B_i^T & 0
\end{matrix} \biggr)$$
where $B_i$ are real matrices and $i=1,2,\ldots,N$, how to prove the following?
$$\det \big( I + e^...
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0
answers
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Is there a relation between covariance matrices and real and imaginary part of eigenvectors?
Apology if my question not clear or appropriate. Consider a complex positive definite sample covariance matrix (SCM) A, generated by a band limited signal on a set of sensors which is termed as data ...
1
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1
answer
333
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How to characterize a linearly-constrained subspace in a projection [closed]
I hope this one is easy. Suppose I have an underdetermined, rectangular matrix $A$ and vector $b$. I want to reason about the subspace where $Ax = b$ and specifically the projection $y:= Tx$. Is there ...
4
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0
answers
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What is the complexity of intersecting two matrix algebras over a finite field?
The following question arose in a joint project with Arkadius Kalka and Adi Ben-Zvi.
Let $\mathbb{F}$ be a finite field, and $M_n(\mathbb{F})$ be the $n\times n$ matrices over $\mathbb{F}$.
For a ...
5
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3
answers
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Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
19
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1
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Number of matrices with given Smith normal form
Denote with $\mathcal{M}$ the set of $(m \times n)$-matrices with integer coefficients bounded by some $K$. Given a matrix $B \in \mathcal{M}$ that is in Smith normal form, is anything known about the ...
2
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0
answers
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Do copairings provide dualities in derived categories?
Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...
1
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0
answers
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Lattice-isotopic essentialization of arrangements
I'm working on a problem related to
$\textbf{Randell's isotopy theorem}$ for complex hyperplane arrangements. I have a question which seems quite obvious. However, I haven't found a rigorous proof ...
2
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0
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A tensor equation related to an invariant of a diffeomorphism
Let $M$ be an $n$-dimensional differentiable manifold, $f : U
\rightarrow V$ a diffeomorphism between open neighbourhoods $U$, $V$
of $M$ with $f(x)=x$ for some $x \in U$, and let $R$, $S$, $T$ be
...
1
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0
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98
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LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]
For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
3
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1
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Create matrix containing values in [0,1] where sum of all diagonals and anti-diagonals is fixed
The problem I am facing sounds at first glance pretty simple. However, as very often, it seems more complicated than I first assumed:
I want to calculate a matrix $P = (p_{j,k}) \in \mathbb{R}^{n \...
12
votes
2
answers
745
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Determinant of a checkerboard Hankel matrix with Catalan numbers
My goal is to compute
\begin{equation}
I = \det \left(\mathbf{I} + \mathbf{A}\right)
\end{equation}
where $\mathbf{A}$ is a $n \times n$ checkerboard matrix filled with Catalan numbers:
$$
\left\{
...
1
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0
answers
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Is my particular finite dimension Toeplitz matrix always strictly positive?
Let $H(\omega), \; -\pi \leq \omega \leq \pi$ be a real-valued function with a continuous band of zeros, that is (for simplicity) $H(\omega)=0, \; |\omega|\geq \beta \pi$.
Define a sequence of banded ...
3
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1
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Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem
T. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where he states as ...
3
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2
answers
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Is the exterior power of a primitive matrix still primitive?
the question is already in the title. Here some more details.
I have a primitive matrix $M$ (primitive means $\exists k\geq 0$ such that $M^k > 0$). I take exterior powers $\wedge^n M$ and I would ...
2
votes
1
answer
719
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Lebesgue measure of set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ are linearly dependent
I've asked this question here on math.stackexchange, but I have been unable to solve this yet, so I'm hoping I can get some advice here.
Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ ...
9
votes
1
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Samuel Karlin's problem: Probability of positive solution to system of random linear equations
I came to know this problem from Dr. W. Bryc's slides (at University of Cincinnati), and I have been continually working on this problem for almost 5 days using different techniques. But I am only ...
1
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1
answer
131
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Self adjoint operator and vertex conditions in quantum graphs
Let $\Gamma$ be a metric graph with finitely many edges. Consider the operator H acting as $\frac{-d^2}{dx_e^2}$ on each edge $e$, with the domain consisting of functions that belong to $H^2(e)$ on ...
16
votes
1
answer
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How many values determine a norm?
It is well known that for a bilinear form over an n-dimensional vector space, $n^2$ values (on all pairs of basis-vectors) determine it uniquely.
How many values do we need to specify in order to ...
23
votes
4
answers
1k
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Dividing by two in the category of vector spaces
Does every invertible linear map $M$ between $V \oplus V$ and $W \oplus W$ naturally yield an invertible linear map $L$ between $V$ and $W$?
Here "naturally" means "in an $GL(V) \times GL(W)$-...
0
votes
2
answers
108
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Symmetric matrix from a nonsymmetricc matrix
Basically this is a part of a long algorithm to calculate some matrix properties.
Given an upper triangular square matrix R, how can I find an orthonormal matrix W (possibly iteratively) such that WR ...
1
vote
0
answers
567
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Bounding the largest Singular value
D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$.
B is any $n \times n$ n.n.d. matrix.
What will be the sharpest upper bound on the largest eigenvalue of:
$(D+B)^{-1}D^2(...
2
votes
0
answers
249
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Smallest Singular Value of a Random Matrix with Dependent Entries
Overview
I am trying to bound from below the smallest singular value $\sigma_{n}$ of a sequence of symmetric $n$ by $n$ random matrices $M_{n}$ with dependant entries. In particular, I would like to ...
1
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0
answers
449
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Complex Hessian Signature
It' all, simply, about the signature of a matrix.
Let $\Omega\subseteq\Bbb C^n$ open, $r:\Omega\to\Bbb R$ twice differentiable (real differentiable, not necessarely complex differentiable, i.e. not ...
3
votes
1
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Find a line such that sum of perpendicular distances of points to the line is minimized
Given a set of points (column vectors) $S = \{p_1, p_2, \cdots, p_n\} \subset \Re^d$, let $A \in \Re^{n \times d}$ be a matrix of which each row is just $p_i^T$. It is easy to find a unit vector $s_1$ ...
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1
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About expectation norms on graphs
Let $S \subseteq V$ of a $d-$regular graph $G$ such that $\mu = \frac{\vert S \vert }{\vert V \vert } $. Let $A$ be the adjacency matrix of the graph. Then define the quantity $\phi(S)= \frac{E(S,\bar{...
1
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2
answers
276
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Solving $Ax=e_k$ for standard basis vector $e_k$, sparse $A$
Given a sparse matrix $A \in \mathbb{R}^{n \times m}$, are there any efficient methods for determining whether there exists an $x \in \mathbb{R}^m$ such that
$Ax=e_k$,
the $k^{th}$ standard basis ...
1
vote
0
answers
89
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Volume under the intersection of scaled simplices
This is rather specific but I need to compute the volume under the intersection of rescaled simplexes, that is, the volume of the space:
$\left\{x \in \mathbb{R}^n|\sum_i c_{ki} |x_i| \leq1\; k = 1 \...
0
votes
1
answer
122
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Minimal dimension of a Lie algebra of matrices, with a restrictive property
Let $\mathfrak{g}$ be a sub-Lie-algebra of $\mathfrak{gl}_n(\mathbb{C})$, the Lie algebra of complex $n\times n$ square matrices.
Let us call $(H)$ the hypothesis: for all $x, y\in\mathbb{C}^n$, ...
1
vote
0
answers
111
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Notions of consistency / heterogeneity in sets of vector values?
The problem
Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1):
$$u=(u_1 \cdots u_n), n\in\mathbb{N}$$
$$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$
I would ...
1
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1
answer
260
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Dense symmetric unitary integer matrix?
Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
0
votes
0
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594
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Hadamard / matrix product adjoint
First of all I would like to thank everyone over here at mathoverflow for their amazing generosity and help (for both pros and newbies like myself).
I apologize if this question seems dumb; I'm a new ...
3
votes
1
answer
857
views
Extension of scalars and projective limits
Consider a morphism of commutative rings $h\colon R\rightarrow S$. This gives rise to a functor $h^*\colon{\sf Mod}(R)\rightarrow{\sf Mod}(S)$, called scalar extension by means of $h$. This functor ...
9
votes
5
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The maximal eigenvalue of a symmetric Toeplitz matrix
Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$.
Is there any ...
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0
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Showing a wedge product is nonzero
Let $V$ be a complex vector space of dimension $n$, equipped with a Hermitian inner product whose Kahler form we denote by $\omega$. Let's set $P = \bigwedge^{2p} V^*$ and $Q = \bigwedge^{2q} V^*$ for ...
3
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2
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Expectation of Gaussian random vector & arbitrary function thereof?
I saw in a paper (https://www.princeton.edu/~wbialek/rome/refs/bialek+ruyter_05.pdf Eq.37) the following identity:
where the <.> operator refers to a population average.
No source or ...
7
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0
answers
334
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Does this inequality always hold?
Denote the adjacency matrix of a given undirected graph by $g$. It is an $n$-by-$n$ symmetric Boolean matrix with elements on the diagonal to be zero ($n\geq 3$). Let $g_{12}=g_{21}=g_{13}=g_{31}=1$ ...
6
votes
1
answer
855
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Geometric interpretation for partial trace?
This MO question asks for a geometric interpretation of the trace of a linear transformation. I'm wondering about a geometric interpretation of partial trace.
Given a linear transformation $f: X\...
4
votes
1
answer
465
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Matrix Elements of Real Representations
I asked this question over at Math.StackExchange and despite having had a bounty on it I did not receive an answer.
Suppose that $G$ is a finite group and we have a unitary irreducible representation ...
2
votes
1
answer
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About distinct eigenvalues of a graph
if a graph with adjacency matrix $A$ and Laplacian $L$ has $k$ distinct eigenvalues then does this fact naturally help define or prove existence of a polynomial $p$ of degree $k-1$ such that $[p(A)]_{...
3
votes
1
answer
544
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Reducing eigenvalues of symmetric PSD matrix towards 0: effect on ratios of original matrix elements?
Let $\boldsymbol{S}$ be $k \times k$ positive semi-definite real symmetric matrix with eigen decomposition $\boldsymbol{S} = \boldsymbol{X} \boldsymbol{\Lambda} \boldsymbol{X}'$ ($\boldsymbol{\Lambda}$...
5
votes
1
answer
229
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Dimensions of a vector space akin to modular symbols
The group $\operatorname{SL}_2(\mathbb Z)$ acts on polynomials in two variables $\mathbb C[x,y]$ via $A\cdot f(x,y)\mapsto f(A^{-1}.(x,y))$ where $(x,y)$ is regarded as a column vector. There are two ...
9
votes
1
answer
379
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Bound on the ratio of top 2 eigenvalues
Let $P$ be a $n \times n$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = 1 - (n-1)\tau$ where $0<\tau < \frac{1}{n}$.
It is clear that the largest eigenvalue of $P$ is 1, ...
2
votes
2
answers
194
views
Matrix inequality
Let $\mathbf{Z,R}$ two Hermitian semidefinite positive matrices with all eigenvalues larger than one. Intuition drives me that
$\mathbf{R}^{-1/2}\mathbf{Z} \left(\mathbf{R}^{-1/2}\right)^H - \...
5
votes
0
answers
246
views
A weak Perron-Frobenius property for sets of positive matrices
A popular form of the Perron-Frobenius theorem states the following result: if $A$ is a $d \times d$ real matrix all of whose entries are positive, then the spectral radius of $A$ is a simple ...