# Questions tagged [linear-algebra]

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

**13**

votes

**0**answers

289 views

### Cardinality vs. isomorphism type of vector spaces without choice

One of the classical uses of the existence of bases of vector spaces (which is equivalent to the axiom of choice) is the following theorem:
If $V$ is an infinite vector space over a field $F$, and $...

**2**

votes

**0**answers

170 views

### A parametrization of stable matrices

Let $A\in\mathbb{R}^{n\times n}$ be a diagonalizable matrix with real and strictly negative eigenvalues. Furthermore, suppose that $\mathrm{tr}(A)=-1$.
My question. I'm wondering whether it is ...

**-1**

votes

**1**answer

70 views

### How many hyper-rectangle-like objects are intersecting a hyperplane?

Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$.
Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set).
Is $\ \#\Big\{H\in\big\{ \{-1\},...

**4**

votes

**1**answer

163 views

### higher Casimirs for $\mathfrak{sl}$

The Wikipedia universal enveloping algebra suggest a way to obtain higher Casimir operators (e.g. generators of the center of $\mathfrak{U(g)}$ for $\mathfrak{g}$ semisimple) by evaluating certain ...

**4**

votes

**1**answer

160 views

### 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 ...

**7**

votes

**1**answer

273 views

### Block matrices and their determinants

For $n\in\Bbb{N}$, define three matrices $A_n(x,y), B_n$ and $M_n$ as follows:
(a) the $n\times n$ tridiagonal matrix $A_n(x,y)$ with main diagonal all $y$'s, superdiagonal all $x$'s and subdiagonal ...

**2**

votes

**0**answers

146 views

### 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-...

**7**

votes

**1**answer

396 views

### 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 $A^-_p$ ...

**3**

votes

**1**answer

138 views

### 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(...

**13**

votes

**0**answers

277 views

### 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&...

**2**

votes

**0**answers

85 views

### 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 ...

**13**

votes

**3**answers

777 views

### 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 ...

**5**

votes

**0**answers

160 views

### 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 } &...

**4**

votes

**2**answers

246 views

### 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{...

**1**

vote

**2**answers

104 views

### 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 ...

**6**

votes

**1**answer

221 views

### Which zero-diagonal matrices contain the all-one vector in their columns' conic hull?

Let $A$ be a non-negative zero-diagonal invertible matrix. Which $A$ make the following assertions true, which are all equivalent:
The all-one vector $j$ is contained in the conic hull of $col(A)$.
...

**2**

votes

**1**answer

76 views

### 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=[...

**1**

vote

**0**answers

62 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 ...

**2**

votes

**0**answers

54 views

### 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 ...

**9**

votes

**2**answers

392 views

### 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 ...

**5**

votes

**1**answer

233 views

### 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 \\
...

**4**

votes

**3**answers

462 views

### Non linear matrix equation

I want to solve the following non linear matrix equation for $X\in\mathbb{R}^{N\times N}$:
\begin{equation}
XX^{\top}+ABX^{\top}-A=0 \qquad (1)
\end{equation}
For a given matrices $A\in\mathbb{R}^{...

**4**

votes

**2**answers

144 views

### Hyperrectangle that contains most of cube's interior (except its vertices)

Let $n>0$, and let $p,q\in (0,1)$ such that $p<q$.
Is there a hyperrectangle $H$ that satisfies the following:
$\forall i\in{1,\dots,n}:\\ H\supset \prod_{j=1,\dots,n}
\begin{cases}
[p,q], &...

**11**

votes

**0**answers

191 views

### 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 $...

**3**

votes

**2**answers

122 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,...

**0**

votes

**1**answer

79 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!

**3**

votes

**0**answers

105 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 $M_n$?

**3**

votes

**1**answer

107 views

### 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 ...

**6**

votes

**1**answer

118 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={\...

**2**

votes

**0**answers

56 views

### Recovering a rank-one matrix from its eigendecomposition after randomized rounding

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{ ...

**3**

votes

**0**answers

78 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(...

**3**

votes

**0**answers

49 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}...

**3**

votes

**2**answers

184 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 ...

**2**

votes

**0**answers

55 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 \...

**1**

vote

**0**answers

100 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)...

**9**

votes

**0**answers

160 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
...

**1**

vote

**0**answers

59 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, ...

**0**

votes

**1**answer

80 views

### Degree of roots of unity in the spectrum of an integer matrix

Let $A$ be an $n\times n$ integer matrix whose all eigenvalues are roots of unity. It is known that for $n=2$, the degrees of these roots can be $1,2,3,4$, or $6$.
What are the degrees for arbitary $...

**0**

votes

**0**answers

83 views

### Matrices with half columns nonnegative and half columns nonpositive real numbers

Suppose I want to study square even dimensional matrices that have the following property: Half of the columns have nonnegative entries and the other half have nonpositive entries. Now given an ...

**1**

vote

**1**answer

104 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 $B=A^k$ ...

**4**

votes

**1**answer

141 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 ...

**0**

votes

**1**answer

66 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 $$\left\{ K_{i}\right\} _{i=1}^{M}, U_{i}D_{i}U_{i}^{T}=K_{i},\quad i=1,\dots,M,$$ ...

**17**

votes

**2**answers

423 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 ...

**1**

vote

**0**answers

53 views

### Linear dependence of solution?

Consider the function
$f_k(c):=\sum_{n=0}^{\infty} c^{n^k}$ where $k\ge 1$ is an integer. This one obviously converges for $\left\lvert c \right\rvert <1.$
In the following we want to study the ...

**1**

vote

**1**answer

117 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**

votes

**3**answers

207 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 ...

**3**

votes

**1**answer

158 views

### 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 ...

**-2**

votes

**1**answer

100 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 $A$?

**5**

votes

**2**answers

135 views

### 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 $A \in W^{1,p}(\Omega;\text{End}(\mathbb{R}^n)) \cap C(\Omega;\text{End}(\mathbb{R}^n))$ ...

**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 ...