# Questions tagged [linear-algebra]

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

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61 views

### Show the spectral radius of a matrix is smaller than 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...

**0**

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33 views

### Find the analytical form for the spectral radius of a special sparse matrix, or its order approaching 1

Let $\hat{\bf H}$ be a $p\hat{N}\times p \hat{N}$ sparse matrix consisting of $p\times p$ blocks, where each block is of size $\hat{N}\times\hat{N}$. The values in $\hat{\bf H}$ is illustrated below (...

**1**

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**0**answers

57 views

### When are “square spans” not transversal?

Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\...

**2**

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33 views

### Volume interpretation of number of perfect matchings in bipartite planar graphs

Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...

**0**

votes

**1**answer

103 views

### Smallest collection of linear operators satisfying isometry property

Let $\mathscr{A}=\{(A_{1,1},A_{1,2},A_{1,3}),...,(A_{S,1},A_{S,2},A_{S,3})\}$ be a collection of linear operators $A_{n,k}:\mathbb{R}^2 \rightarrow \mathbb{R}^2$. For $u$ and $v\in (\mathbb{R}^2)^3$, ...

**6**

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159 views

### Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller ...

**2**

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**0**answers

69 views

### How to formally connect the log-Euclidean and Riemannian metrics for Symmetric Positive Definite matrices?

I'm a statistician working on a research project dealing with metrics on SPD matrices, specifically the log-Euclidean $d_{LE}(X, Y) = \|\log(X) - \log(Y)\|$ and the Riemannian metric $d_{R}(X, Y) = \|\...

**1**

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**1**answer

78 views

### A question on a special “metric”

Suppose we have a function $F: [a,b]^n \to \mathcal{M}_{n \times n }(\mathbb{R})$ where $\mathcal{M}_{n \times n }(\mathbb{R})$ is the space of $n \times n$ real matrices, a compact set $B \subset \...

**3**

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**1**answer

77 views

### Largest eigenvalue of product of orthogonal-projection rank-1 perturbation

Suppose I have a symmetric positive definite matrix $A \in \mathbb{R}^{n \times n}$ with $n$ linearly indepedent columns $a_1,...a_n$ in $\mathbb{R}^n$. All columns $a_i$ has norm 1, but they are not ...

**4**

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**1**answer

332 views

### Subsets $E$ of $\mathbb{F}_{p^k}$ with vanishing polynomial subset sums

The following question arose in some discussions recently as a misunderstanding of another problem.
Question: Which subsets $E\subset \mathbb{F}_{p^k}$ satisfy the property that $ \sum\limits_{x\in E}...

**2**

votes

**1**answer

122 views

### Matrices with distinct columns

Let $M$ be an $n \times m$ matrix over $\mathbb{F}_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$: i.e., it is possible to have a matrix with fewer rows that still has $m$ unique ...

**5**

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**1**answer

230 views

### Lefschetz operator

Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2\mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...

**1**

vote

**1**answer

103 views

### Set of integer matrices $A$ such that $(A^k)_{k\in\mathbb{N}}$ is eventually periodic

This is a more elegant version of the original version which can be found below; it is based on a suggestion by Peter Mueller.
Let $\mathbb{N}$ denote the set of positive integers and for $n\in\...

**2**

votes

**1**answer

179 views

### $2$-adic valuation on $\mathbb Q (\sqrt{-15})$

I was reading the book "The structure of groups of prime power order". In the book (page 226, Example 10.1.18) it is proved that
$-15$ has a square root is $\mathbb Z_2$ where $\mathbb Z_2$ denotes ...

**6**

votes

**1**answer

204 views

### Covering the finite plane with lines

This is, essentially, a geometrically rendered version of the question I asked a week ago, with the emphases slightly shifted; it seems more natural and appealing (to me, at least) in this form.
Let ...

**1**

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**1**answer

230 views

### How to verify the characteristic polynomial? [closed]

I am computing the characteristic polynomial of a matrix over a number field, using the minimal polynomial of it. Is there a fast way to verify the characteristic polynomial of a big matrix ?

**4**

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176 views

### Forcing scalar products to avoid prescribed values

Let $p$ be a prime, and $n\ge 1$ an integer number. Suppose that the (not necessarily distinct) vectors $v_1,\dotsc,v_N \in{\mathbb F}_p^n$ satisfy the following condition:
\begin{gather}
\text{For ...

**1**

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**0**answers

74 views

### Eigenvalues of non-negative block matrices

$B$ is a non-negative irreducible block matrix as follows:
$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...

**2**

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**0**answers

112 views

### Maximal number of $S_n$-conjugates living in a hyperplane

Let $v=(a_1,\dots,a_n)\in\mathbb{R}^n$ where the $a_i$ are distinct and positive. For $\sigma\in S_n$, let $\sigma(v)=(a_{\sigma(1)},\dots,a_{\sigma(n)})$. For any hyperplane $H$ through the origin, ...

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78 views

### Generic deformation of matrix

Let $A(x)$ be a $m \times n$ matrix, whose entries are real polynomials $f_{i,j}:\mathbb{R}^S \to \mathbb{R}$. Denote the ith row by $f_i$ And let $rk:\mathbb{R}^S \to \mathbb{N}$ be the rank function ...

**2**

votes

**1**answer

145 views

### Linear subspaces in quadric hypersurfaces

Consider $H_1,H_2,H_3\subset\mathbb{P}^{2m+1}$ three general linear subspaces of projective dimension $m$.
Then there exists a quadric hypersurface $Q^{2m}\subset\mathbb{P}^{2m+1}$ containing $H_1,...

**4**

votes

**1**answer

75 views

### Separate the trivial partition by a linear hyperspace

Let $e=[1,1,\ldots,1]\in\mathbb{Z}^n$. I am looking for a way to find a vector $a\in\mathbb{Z}^n$ such that:
$\langle a,e\rangle=0$ and
for every nonnegative $v\in\mathbb{Z}^n$ such that $\langle e,v\...

**2**

votes

**0**answers

44 views

### Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

This is cross post to the question at MSE.
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 ...

**11**

votes

**1**answer

314 views

### Decide if a matrix is transposable

A matrix $M$ is called transposable if it can be transformed into its transpose $M^t$ via row and column permutations.
Is there an efficient a way/algorithm to decide if a given matrix is
...

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26 views

### Approximate Simultaneous Diagonalization of Non-Hermitian Matrices

Let $A_1,A_2$ two $n\times n$ complex matrices. $A_1$ and $A_2$ are also non-normal, especially, non-hermitian and do not commute. I would like to find an invertible matrix $V$ such that
$$
\sum_{i=1,...

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**3**answers

205 views

### Determinant of a block matrix with many $-1$'s

For an array $(n_1,...,n_k)$ of non-negative integers and non-zero reals $a_1,...,a_k$, define a block matrix $M$ of size $n=n_1+\cdots+n_k$ as follows:
The main diagonal has blocks of sizes $n_i$ and ...

**1**

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33 views

### Choice of residual function for least squares error minimization

Good morning,
I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data.
I want to determine two parameters $K_{IC}$, $C_f$ in the basic equation given as
$K_{IC} = \sigma \sqrt{D} k_0(\...

**20**

votes

**2**answers

445 views

### $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class

It is easy to classify conjugacy classes in $GL_n(\mathbb Q_p)$ by linear algebra. How to classify $GL_n(\Bbb Z_p)$ conjugacy classes in a $GL_n(\Bbb Q_p)$ conjugacy class? For example, for general ...

**4**

votes

**2**answers

171 views

### The maximal size of intersection of two sets

Let $S=\{1,2,\cdots,2n\}$, and $S_i \subseteq S(i=1,2,\cdots,n+1)$ be $n+1$ subsets, each of which contains half of the $2n$ elements, namely $|S_i|=n$. Consider the following expression:
$$M=\max_{1\...

**3**

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**0**answers

194 views

### Matrix Inequality: Traces of $n$th powers

Let $A, B$ be matrices over $\mathbb{C}$ of the same dimensions (not necessarily square). With $'$ denoting conjugate-transpose, and tr the trace, show for $n\in\mathbb{N}$ that
$ 2\,\mathrm{Re}\, \...

**0**

votes

**1**answer

97 views

### About roots of the equation $det(A-\lambda B)=0$ [closed]

Let $A,B$ be square real symmetric matrics of the same degree $n \geq 3$ without common isotropics vectors (i.e. there is no a nonzero vector $x\in \mathbb R^n$ such that $x^TAx=x^TBx=0$).
Are roots ...

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77 views

### Matrix completion in $2\times2$ case by nuclear norm minimization to guarantee rank $1$?

Does fixing diagonal entries and minimizing nuclear norm under weighted sum of entries conditions produce a rank $1$ matrix? I think the answer for this is no.
At least could it be true in $2\times2$ ...

**6**

votes

**1**answer

141 views

### Bounding the eigenvalues of $B A B^T$ with the eigenvalues of $A$

Given a Hermitian positive semi-definite $n \times n$ matrix $A$ and a rectangular $m \times n$ matrix $B$, is there anything that can be said about the eigenvalues of the matrix $B A B^T$?
It seems ...

**2**

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**0**answers

65 views

### What is the most efficient path for a robot without turning radius?

I recently programmed a most efficient path for a robot going from $(x_1, y_1 , \theta_1)$ to $(x_2, y_2)$. The bot does a combination of turning and moving at any given point, based on the difference ...

**0**

votes

**1**answer

256 views

### Conjecture that relates matrix systems with some specific functions as solution sets

what is written below is a conjecture that I posed , and I ask for a proof or a disproof of it .I have checked the conjecture from $n$=$1$ up to $n$=$10$ using Matlab, and all results were in ...

**8**

votes

**1**answer

181 views

### Making binary matrix positive semidefinite by switching signs

Let $A \in \{\pm 1\}^{n \times n}$ be a symmetric matrix whose diagonal entries are $+1$. Let $f(A)$ be the smallest number of signs we need to change in $A$ so that it becomes positive semidefinite (...

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**2**answers

181 views

### Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...

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**2**answers

171 views

### Degree of the variety of independent matrices of rank $\leq r$?

Consider an $m$-by-$n$ matrix $A$ with entries in a field $k$; we can see $A$ as a point in the affine space $\mathbb{A}^{m n}$. The rank of $A$ will be $\leq r$ (where $r<\min(m,n)$) if and only ...

**0**

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53 views

### Can a rational symmetric matrix $A$ be diagonalized as $A = P \Lambda S$ for some $P, S $ in the general linear group?

Let $A$ be a $3 \times 3$ symmetric matrix with rational entries. Does there exists a unique pair of matrices $P, S \in \text{GL}_3(\mathbb{Z} )$ depending on $A$ such that $A = P \Lambda S$, where $\...

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30 views

### Defining a notion of “volume of its lattice” for non-rational subspaces

Let $V\subseteq \Bbb R^n$ be a vector subspace. If $V$ is rational, i.e. has a basis consisting of elements in $\Bbb Z^n$, then there’s a well-defined notion of the “volume of the lattice of V”:
$$\...

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vote

**2**answers

154 views

### How to compute inverse of sum of a unitary matrix and a full rank diagonal matrix?

$C = A+D$, $A$ being a unitary matrix and $D$ a full rank diagonal matrix. Is there any easy way to compute $C^{-1}$ from $A^{-1}$ and $D$, if it exists?
I am interested in this question, because my ...

**2**

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**1**answer

144 views

### The effect of random projections on matrices

Let $A\in\mathbb{R}^{n\times n}$ be a given normal matrix, i.e. $A^TA=AA^T$. Let $P_s\in\mathbb{R}^n$ be a random projection matrix to an $s$-dimensional subspace in $\mathbb{R}^n$.
Suppose $\frac{A+...

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votes

**2**answers

453 views

### Reference request: Oldest linear algebra books with exercises?

Inspired by the recent success of my "soft question" here, I also have to ask, what are some of the oldest linear algebra books out there with exercises? I'm fine with or without solutions, either way....

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**2**answers

248 views

### How do fractional tensor products work?

[I asked and bountied this question on Math SE, where it got several upvotes and a comment suggesting it was research-level, but no answers. So I'm reposting here with slight edits, but please feel ...

**2**

votes

**1**answer

232 views

### A peculiar operation on $M_2(\mathbb Z)$ which along with the usual matrix addition, makes $M_2(\mathbb Z)$ into a commutative ring with unity [closed]

For $A=\begin{pmatrix} a_1 & b_1 \\ c_1&d_1 \end{pmatrix}, B=\begin{pmatrix} a_2 & b_2 \\ c_2&d_2 \end{pmatrix}\in M_2(\mathbb Z)$, define
$A*B:=a_1L_1BR_1+b_1L_1BR_2+c_1L_2BR_1+...

**1**

vote

**1**answer

49 views

### Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?

This is a cross-post to the question I asked at MSE.
The set of Schur stable matrices is
\begin{align*}
\mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\},
\end{align*}
where $\rho(\cdot)$ denotes ...

**2**

votes

**1**answer

141 views

### Irreducible but not completely irreducible

Consider a product of i.i.d. $d\times d$ random matrices $A_{i}$
(with $\mathbb{E}\log\left\Vert A_{i}\right\Vert <\infty$).
Let $F:M\times \mathbb R^d\to M\times \mathbb R^d$ be a linear cocycle, ...

**3**

votes

**2**answers

102 views

### Eigenvalue density of a symmetric tridiagonal matrix

Let $A_n\in\mathbb{R}^{n\times n}$ be defined as
$$
A_n=\begin{bmatrix} a & b & 0 & \cdots & \cdots & 0 & 0\\ b & a & b & \cdots & \cdots & 0 & 0\\ 0 &...

**1**

vote

**1**answer

81 views

### Can eigenvectors determine their original matrices given the basis of matrices space is small?

Let $A_1$, $A_2$, and $A_3$ be three different mutually orthogonal random hermitian operators, say dimension of $30\times 30$. For three random real numbers $a_1$, $a_2$, $a_3$, we have
$$
(a_1A_1+...

**1**

vote

**1**answer

37 views

### Marginal stability of discrete linear time-invariant system

I have a question about marginal stability of a system:
\begin{equation}
\mathbf{x}[k] = \mathbf{A}\mathbf{x}[k-1]
\end{equation}
I would adapt the definition of marginal stability from this ...