Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,679
questions
7
votes
1
answer
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A new determinant question for primes $p\equiv3\pmod4$
Let $p$ be an odd prime, and let $(\frac{\cdot}p)$ denote the Legendre symbol. Motivated by my question http://mathoverflow.net/questions/310301, here I introduce the matrices $A^+_p$ and $A^-_p$ ...
- 85.1k
9
votes
2
answers
641
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A trace-constrained maximization problem in the cone of positive definite matrices
Let $A\in\mathbb{R}^{n\times n}$ be a matrix having eigenvalues with strictly negative real part (in other words, $A$ is supposed to be Hurwitz stable). Let $\mathrm{tr}(\cdot)$ denote the trace ...
- 2,682
4
votes
1
answer
840
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Which inner products preserve positive correlation?
Suppose we have a symmetric PD or PSD matrix M which induces an inner product $\langle \cdot, \cdot \rangle_M$. If we have that $\langle x, y \rangle > 0$ for two unit vectors $x$, $y$, are there ...
- 19.4k
3
votes
1
answer
301
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Reference on completely positive maps which are isometries
Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
- 3,954
2
votes
0
answers
182
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Evaluate a curious determinant with Legendre symbol entries
Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendre symbol. R. Chapman's conjecture on the exact value of the determinant of
$$C_p:=\left[\left(\frac{i-j}p\right)\right]_{0\le i,j\le (p-...
- 14.5k
3
votes
1
answer
174
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Is the convergence of $\dot{x}=2A(t)x$ faster than that of $\dot{x}=A(t)x$?
Let $x \in \mathbb{R}^{n}$ and $A(t) \in \mathbb{R}^{n\times n}$. If $\dot{x}=A(t)x$ and $\dot{x}=cA(t)x$ with $c>1$ are exponentially stable. Is the convergence rate of $x$ to zero of $\dot{x}=cA(...
- 16.3k
2
votes
0
answers
228
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An inequality concerning the solution of a Lyapunov equation
Let $A$ be an Hurwitz stable matrix (i.e. the real part of the eigenvalues of $A$ lie in the left-half plane) and $Q$ be a positive semidefinite matrix ($Q\ge 0$, for short). Let $P>0$ be the ...
- 2,682
13
votes
0
answers
345
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A determinant problem for primes $p\equiv 1\pmod4$
Let $p$ be an odd prime, and let $A_p$ denote the matrix
$$[a_{ij}]_{1\le i,j\le (p-1)/2},$$
where
$$a_{1j}=\left(\frac jp\right),\ \ \text{and}\ \ a_{ij}=\left(\frac{i^2+j^2}p\right)\ \text{for}\ i&...
- 14.5k
13
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3
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Is $-\det\big[\big(\frac{i^2+j^2}p\big)\big]_{1\le i,j\le (p-1)/2}$ always a square for each prime $p\equiv 3\pmod 4$?
Let $p$ be an odd prime and let $S_p$ denote the determinant
$$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$
with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my ...
- 30.7k
4
votes
2
answers
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Fast projection onto a subspace
Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{...
- 12.3k
2
votes
2
answers
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fast way to calculate normal to set of vectors with $\pm$1 entries
Say I have a set of $(n-1)$ linearly independent vectors $\mathbf{v}_i$ of dimension $n$ with entries $\pm1$. I am interested in finding the $n-$dimensional vector $\mathbf{u} $which is normal to the ...
- 25.5k
3
votes
1
answer
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What can be said about the relationship between the eigenvalues of a negative definite matrix and of its Schur complement?
I have two problems related to eigenvalues of negative definite matrices:
I have a matrix $M\prec0$ (symmetric and all eigenvalues are negative) and $S=M_{11}-M_{12}M_{22}^{-1}M_{21}$ by taking $M=[...
6
votes
0
answers
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Determinant arising in a problem from probability
Consider the determinant:
$$\Delta:=
\left|\begin{array}{cccc}
A_{j_1} & A_{k_1} & A_{j_1}A_{k_1} & 1 \\
A_{j_2} & A_{k_2} & A_{j_2}A_{k_2} & 1 \\
A_{j_3 } & A_{k_3 } &...
- 3,507
11
votes
0
answers
225
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Matrices that admit a power that is symmetric
We fix an integer $n\geq 2$. Let $S_n$ be the set of real symmetric matrices in $M_n(\mathbb{R})$. We consider the algebraic sets $Y_k=\{A\in M_n(\mathbb{R});A^k\in S_n\},k\geq 2$ and the sequence $...
1
vote
0
answers
241
views
Determinant and restriction of scalar
Let $E/F$ be a finite separable extension of fields, and $V$ a finite dimensional vector space over $E$. Let $T\in\operatorname{End}_EV$ be a linear operator on $V$, and let $\det(T)$ be its ...
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votes
0
answers
182
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Can projecting a simplex onto orthogonal subspaces exposes the same vertices and edges?
Given the regular $n$-dimensional simplex $S\subset\Bbb R^n$ with $n\ge 4$, as well as two orthogonal subspaces $V,W\subset\Bbb R^n$ of dimension $\ge2$ (not necessarily of same dimension, not ...
- 12.6k
5
votes
1
answer
584
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Largest Eigenvalue of a Matrix with Special Form in terms of n
In one step of solving a difficult problem, I would like to know the largest eigenvalue of a matrix with this pattern:
$$A_n = \begin{bmatrix}
0 & 0 & 0 & 0 &\dots & 0 \\
...
- 103
3
votes
2
answers
144
views
Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$
I am working in data science and I have to deal with the following problem for which I would like to find a simplification:
We call a function almost positive if $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,...
- 489
2
votes
0
answers
92
views
Recovering a rank-one matrix from its eigendecomposition after randomized rounding [closed]
Let $A = xy^T$ be a rank-$1$ matrix, and suppose every entry of $A$ is in $[0,1]$. We can create a binary matrix $A_{\rm rounded}$ by setting
$$ [A_{\rm rounded}]_{ij} = \begin{cases} 1 & \mbox{ ...
- 121
0
votes
1
answer
8k
views
Product of Positive Matrices
Is the product of non-negative definite matrices also non-negative definite? If not, let A and B be non-negative definite matrices, is '$\operatorname{tr}(A^T B) \ge0$' ?
CommunityBot
- 1
4
votes
1
answer
275
views
Is there a fast algorithm to test positivity of all principal minors of non-symmetric matrix?
I have a matrix $A \in \mathbb{R}^{n \times n}$ with positive eigenvalues. In the symmetric case, Sylvester's criterion implies that all the principal minors are positive. In the non-symmetric case, ...
CommunityBot
- 1
17
votes
2
answers
1k
views
Constructive proof of a rational version of Perron-Frobenius?
In the following, we work with vectors and matrices whose entries are rational numbers. Inequalities between such vectors are understood to be coordinatewise: e.g., two vectors $a = \left(a_1,a_2,\...
- 33.2k
0
votes
1
answer
104
views
Name of a matrix with one column and row removed [closed]
I am looking for the exact name of a matrix where the i-th column and rows have been removed.
I cannot remember how it is called in linear algebra, does anyone got an idea?
Thanks!
- 19.4k
3
votes
0
answers
125
views
An eigenvalue of certain family of matrices
Consider the matrices
$$M_n=\left[\binom{i}j+\binom{2n+1-i}{j-i}+\binom{2n+1-i}j\right]_{i,j=0}^n.$$
I am convinced and hence would like to ask:
Question: Is $0$ an eigenvalue of $M_n$?
- 41.8k
6
votes
1
answer
199
views
Any convergence rule for ${\mathbf X}_k={\mathbf A}{\mathbf X}_{k-1}{\mathbf B}$?
We know iteration ${\mathbf X}_k=\mathbf{A}{\mathbf X}_{k-1}$ converges if the spectral radius of $\mathbf A$ is smaller than 1 (see here). Is there any known rule for iteration ${\mathbf X}_k={\...
- 11.6k
2
votes
1
answer
152
views
Finding a similarities and differences of sent of matrices
Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,
- 21
3
votes
0
answers
208
views
Simultaneous Congruence of Two Matrices
Could you please let me know an answer to the following question on simultaneous congruence of two matrices. This question came up while trying to handle a system of PDE.
QUESTION:
Let $A,B\in M_n(...
- 895
7
votes
4
answers
1k
views
Is the componentwise square-root of a positive-definite matrix also pos.-def.?
Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$ be a matrix with $a_{ij} = a_{ji} \geq 0$ and
$B=(b_{ij})$ with $b_{ij} = \sqrt{a_{ij}}$.
Is $B$ positive-definite whenever $A$ is?
In other words:
$\...
- 33.2k
17
votes
2
answers
506
views
On a special type of normed linear spaces
Let $(V,\|.\|)$ be a normed linear space such that for every group $(G,*)$, every function $f:G \to V$ satisfying
$$
\|f(x*y)\|\ge \|f(x)+f(y)\|,\qquad\forall x,y\in G,\tag{Z}
$$ is a group ...
- 11.4k
4
votes
2
answers
3k
views
Upper bound of spectral radius of the sum of two matrices, one with spectral radius no larger than 1, and the other has small eigenvalues
Suppose I have one $pN\times pN$ matrix $\bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $\bf H$ is in a block-like ...
- 11.6k
2
votes
0
answers
60
views
Linearity of a canonical morphism related to scalar extension and coextension
Let $h\colon R\rightarrow S$ be a morphism of commutative ring. Let $M$ and $N$ be $R$-modules. We consider the canonical morphism of $R$-modules $$p\colon{\rm Hom}_R(S\otimes_RM,N)\rightarrow{\rm Hom}...
- 6,670
3
votes
0
answers
123
views
Number of distinct directions in the set $\mathbb{Z}^2 \cap B(0,R)$
We say that non-zero directions $v, w \in \mathbb{R}^2$ are equivalent if they span the same line (i.e. $\exists C \in \mathbb{R}: v = Cw$.), and distinct otherwise. Given a collection $V \subset \...
- 81
1
vote
0
answers
148
views
Bounds of Procrustes problem
We denote $\|\cdot\|_F$ as the Frobenius norm of some matrix. We define $f: \mathbb{R}^{d\times r}\times\mathbb{R}^{d\times r} \rightarrow \mathbb{R}^{d\times r}$ as the following:
\begin{align}
f(A,B)...
- 33.9k
10
votes
0
answers
228
views
Maximum dimension of a space of $n\times n$ real matrices with at least $k$ nonzero eigenvalues
Let $M_n(\mathbb{R})$ denote the $n^2$-dimensional real vector space
of real $n\times n$ matrices. Let $\rho_k(n)$ denote the maximum
dimension of a subspace $V$ of $M_n(\mathbb{R})$ such that every
...
- 49.5k
1
vote
0
answers
232
views
Gram determinant in dual basis [duplicate]
Assume that $V$ is a $n$ -dimensional inner product space.
Basis $(e_1,...,e_n)$, $(f_1,...,f_n)$ are said to be dual if
$\langle e_i, f_j\rangle=0$ for $i\neq j$, $i,j=1,...,n$ and $\langle e_i, ...
- 19.4k
4
votes
1
answer
186
views
How to find the analytical representation of eigenvalues of the matrix $G$?
I have the following matrix arising when I tried to discretize the Green function， now to show the convergence of my algorithm I need to find the eigenvalues of the matrix $G$ and show it has absolute ...
- 1,583
1
vote
1
answer
123
views
Walks of odd Lengths in a Matrix
Consider the following matrix
$$
A=\left[
\begin {array}{cccc}
1&1&0&0\\ 0&0&1&0\\ 0&0&1&1\\ 1&0&0&0
\end {array}
\right].
$$
Assume that $B=A^k$ ...
- 103k
0
votes
1
answer
464
views
Computing spectrum of convex combination of SPD matrices given individual spectral decompositions
Given the spectral decompositions of a non-commuting collection of symmetric positive definite $N\times N$ matrices $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...
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3
votes
1
answer
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Inertial decomposition of graphs
The problem is this: given a graph $G$, to find a decomposition of $G$, i.e. a set $F$ of vertex-disjoint proper subgraphs of $G$ such that:
$$\text{inertia}(G) = \sum_{H \in F} \operatorname{...
CommunityBot
- 1
4
votes
3
answers
380
views
Is a simple graph the "sum" of a partial order and its dual?
A "$n$-order matrix" $T\in M_n(\mathbb F_2)$ is a matrix such that there exists a partial ordered relation $\leq_T\subset [1,n]^2$ such that :
$T_{ij}=1\Leftrightarrow i\leq_T j$
(where $T_{ij}$ is ...
0
votes
1
answer
291
views
root of identity matrix and lexicographic order
I asked a question here order of a permutation and lexicographic order but it seems*** that a very powerful and rich generalization can be made!
Let $A$ be a finite ring together with an arbitrary ...
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1
vote
1
answer
2k
views
Lanczos algorithm for finding $k$ smallest eigenvector
I am trying (and have been recommended) to use the Lanczos algorithm to find the $k$ smallest eigenvectors. However, all of the literature seems to talk about this algorithm as a way to estimate the $...
- 4,709
4
votes
3
answers
364
views
Extending a continuous map over projective space
Let $X = P^{n-1}(\Bbb C)$ ($(n-1)$-dimensional projective space) with $n \geq 3$, and let $K \subset X$ denote a compact subset. I have a bijective, continuous map $\phi:K \to K$ which satisfies the ...
- 346
3
votes
1
answer
448
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If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [duplicate]
Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
- 95.6k
7
votes
1
answer
331
views
Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?
(Posted this on math.stackexchange and cross.correlated over more than a week ago, but didn't get an answer, and this is a question in my research so this seems like it might have been the better ...
- 178k
18
votes
2
answers
4k
views
Minimum off-diagonal elements of a matrix with fixed eigenvalues
I am an engineer working in radar research. I came accross a problem on which I cannot seem to find literature. I can ask it in two different ways. Perhaps depending on the reader, the alternative ...
-2
votes
1
answer
902
views
What can we say about the rank of the sum of a multiple of the identity matrix and a symmetric rank-$1$ matrix? [closed]
Suppose we have the following symmetric matrix.
$$A = \sigma^2 I + u u^T$$
What can we say about the eigendecomposition of $A$?
17
votes
1
answer
4k
views
Geometric interpretations of matrix inverses
$A$ is an invertible $n \times n$ matrix. Interpret each row of $A$ as a point in $\mathbb{R}^n$; then these $n$ points define a unique hyperplane in $\mathbb{R}^n$ that passes through each point (...
- 4,618
5
votes
2
answers
231
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Can we stay invertible while approximating linear maps in Sobolev spaces?
Let $\Omega \subseteq \mathbb{R}^n$ be an open bounded domain with a smooth boundary. Fix $1<p<n$.
Let $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...
CommunityBot
- 1
18
votes
1
answer
1k
views
A linear algebra problem in positive characteristic
Let $A$ be a symmetric square matrix with entries in $\mathbb{Z}/p\mathbb{Z}$ for a prime $p$ such that all of its diagonal entries are nonzero. Does there exists always a vector $x$ with all ...
- 151k