# Questions tagged [linear-algebra]

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

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### Ref. request: Enumerating elements of Bruhat cells

Given a field $F$ and a natural number $n$, let $B$ be the group of lower triangular, invertible $n \times n$ matrices over $F$. Then
$$GL_n(F) = \biguplus_{\pi \in S_n} B \pi B,$$
where we embed the ...

**2**

votes

**3**answers

278 views

### Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently?
$$(D + ADA^T)^{-1}$$
Hints or references are ...

**7**

votes

**1**answer

138 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 ...

**14**

votes

**3**answers

585 views

### Determining if some permutation of a vector satisfies a system of linear equations

Let $A$ be a matrix and $x$ a fixed vector. How can we determine whether or not there exists a permutation matrix $P$ such that $APx=0$? Does this problem reduce to anything well-understood?

**5**

votes

**1**answer

216 views

### invertible endomorphisms on a space of linear maps between finite-dimensional vector spaces

This is a linear algebra question that came up in my research, and I feel like there ought to be either a simple proof or a simple counterexample, but I have been unable to find either.
Assume $V$ ...

**6**

votes

**2**answers

163 views

### Is there any efficient solution of the matrix equation AXB + (AXB)' + cX = D

I am trying to find the symmetric solution $X\in \mathbb{R}^{p\times p}$ of following matrix equation:
$AXB + (AXB)^T + cX = D$
where $A,B\in \mathbb{R}^{p\times p}$ are symmetric positive ...

**2**

votes

**0**answers

137 views

### When does $S \cap GL_4(\mathbb R)$ have precisely two connected components where $S \subset M_4(\mathbb R)$ is a linear subspace?

This is a cross-post to the question I asked at MSE.
Let $A \in M_4(\mathbb R)$ and $A = (e_2, x, e_4, y)$ where $e_2, e_4$ are standard basis in $\mathbb R^4$ and $x,y$ are undetermined variables. ...

**0**

votes

**0**answers

119 views

### Integral subspace generated by a positive semidefinite matrix

Take $\Sigma $ a real positive semidefinite matrix. Define $P$ to be the smallest projection with the property that for any $\mathbf{a}\in \mathbb{Z}^n$ with $\mathbf{a}^\dagger (I-P)^\dagger \Sigma (...

**0**

votes

**0**answers

24 views

### Simultaneous movement toward barycenters - what can be guaranteed

Suppose a tiling is given in 2D (an embedding of a planar triangulated graph), with all faces convex.
Now suppose one moves each point, one by one, to the barycenter of its neighbors. I think that ...

**4**

votes

**0**answers

160 views

### What is the Jarlskog invariant, conceptually?

Let $U$ be a $3\times 3$ unitary matrix, and call $(u_{ij})$ its coefficients. For $i,j,k,\ell$ in $\{1,2,3\}$ with $i\neq j$ and $k\neq\ell$, consider the quantity:
$$J_{ij,k\ell} := \operatorname{...

**5**

votes

**3**answers

371 views

### Irreducible representations and invariant subspaces

Consider two invertible n-by-n matrices, $n>2$, $X$ and $Y$ over a finite field $k$ (say for simplicity $k=\mathbb Z/ \mathbb Z_2$). Is there any reasonable way to check that there is no proper ...

**1**

vote

**0**answers

120 views

### Sufficient conditions for all eigenvalues simple in stochastic matrix

The "largest" eigenvalue $1$ of a stochastic matrix is well-characterized by the classical Perron-Frobenius theorem. In particular, it gives sufficient conditions for the eigenvalue $1$ to be simple.
...

**2**

votes

**0**answers

163 views

### Intersection of two varieties in $\mathbf{F}_q^n$

Suppose we identify $\mathbf{F}_q^n$ with $\mathbf{F}_{q^n}$. Let $X_n$ be the irreducible hypersurface defined by $Nx=1$ where $N$ is the norm map. There is an analogous hypersurface $X_{n-1}$ in $\...

**0**

votes

**0**answers

127 views

### Closed-form solution about a matrix factorization problem?

I am doubting some equations in paper "Multi-View Learning With Incomplete Views". The paper could be found here. The problem is related to Eq.(2) and Eq.(4b) in this paper. I summarize them here (...

**5**

votes

**1**answer

277 views

### Error in Hoffman-Kunze (normal operators on finite-dimensional inner product space with a cyclic vector)

I'm teaching a second course in advanced linear algebra, following the second half of Hoffman-Kunze. I have come across what I believe to be an error, but I want confirmation (or refutation) by ...

**4**

votes

**1**answer

146 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, ...

**5**

votes

**1**answer

138 views

### Numerical minimization spectral norm under diagonal similarity

This question is a follow up.
Let $A$ be a real square matrix of size $n \times n$. How to determine the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \inf_{D} \lVert D^{-1} A D\...

**0**

votes

**1**answer

173 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 ...

**5**

votes

**2**answers

256 views

### Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e.,
$$
s(A) = \min_{D} \lVert D^{-1} A D\rVert_2,
$$
where $D$ ...

**2**

votes

**1**answer

77 views

### Is a simple graph matrix the sum of a “shiftordered” matrix and its transposed matrix

This is the generalization of a question Is a simple graph the "sum" of a partial order and its dual?
Nik Weaver found a counterexample in a very nice, complete (and instantaneous!) answer,...

**4**

votes

**3**answers

336 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 ...

**2**

votes

**0**answers

123 views

### lower bounding the absolute value of a determinant

In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or ...

**0**

votes

**1**answer

82 views

### How to decompose a matrix into its orthogonal and diagonal parts (assuming it has that form)? [closed]

Assume that $A = U * S$ for $U$ orthogonal and $S$ diagonal, ordered and positive.
If I only know $A$, is it possible to obtain $U$ and $S$?
My first guess would be taking the singular value ...

**0**

votes

**0**answers

187 views

### Are the integers a vector space or algebra over “some” field or over “some” ring?

Every vector $v$ in a finite-dimensional vector space space $V$ of dimension $n$ over a field $F$ has a unique representation in terms of a basis ${\frak B} \subseteq V$, where a basis for $V$ is a ...

**1**

vote

**1**answer

194 views

### Is there a generalization of eigenvalues and eigenvectors to tensors?

Two perhaps ill-posed or just silly questions:
Let $n>0$, $T$ be an $(n+2)$-tensor, and $\otimes$ denote the Kronecker product of tensors. Is there a tensor generalization for the fundamental ...

**2**

votes

**1**answer

86 views

### Local-Global Principle in Graph Spectrum

The question is a bit vague, but any ideas/directions will be appreciated.
Let us fix an $n$-vertex $d$-regular graph $G=(V,E)$. As I understand it, the eigenvalues of the adjacency matrix $A$ of $G$ ...

**4**

votes

**0**answers

72 views

### Monoid cohomology of $\mathbb{N}$ for a linear algebraic group

Let $k$ be a finite field and $k_E:=k((X))$ denote the field of Laurent series over $k$. We define a Frobenius endomorphism on $k_E$ via $f(X)\mapsto f(X^p)$. We choose a lift $\varphi:k_E^{sep}\...

**1**

vote

**1**answer

60 views

### Conditions to obtain a real logarithm of a unitary unimodular complex matrix?

The problem statement is the following:
$$U=\exp\{iV\}$$
where $U$ is a unitary unimodular matrix of the following form:
$$U=\begin{bmatrix}u_1+iu_2&u_3+iu_4\\-u_3+iu_4&u_1-iu_2\end{bmatrix}...

**2**

votes

**1**answer

122 views

### Controlling angles between vectors using sum of subvector angles?

This is a technical question coming out of my research.
Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that
$$
\...

**1**

vote

**0**answers

124 views

### Decomposition of Determinant of Sub-Matrices of a Matrix

Consider an $n \times n$ matrix $\bf A$ over a field. Let $\bf A$ is constructed by the product of $n \times n$ matrices $B_i$, for $1\leq i \leq m$ which means
$$
{\bf A}=\prod_{i=1}^m\, {\bf B}_i\, ...

**1**

vote

**0**answers

77 views

### Decomposition of a Matrix by Sparse Matrices

Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over $\mathbb{F}$.
$\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$ be at most ...

**1**

vote

**0**answers

32 views

### find linear approximation of non-linear matrix transform [closed]

I have a square matrix denoted as $A$ and an element-wise square operator $\sigma$, such that $\sigma(A)=a_{ij}^2$,$\forall i,j$, $a_{ij}$ is the ith row and jth column element of $A$. Is there exists ...

**0**

votes

**0**answers

35 views

### Unstable convergence of a Poisson equation

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum ...

**1**

vote

**1**answer

150 views

### Eigenvalues of n matrices

For $i \in \{1,...,n\}$, let $A_i=[a^i_{jk}]$ be a symmetric matrix.
For $i \in \{1,...,n\}$, we assume that $a^i_{jj}=0$ for all $j$ and rank $(A_i)=2$.
Then it has one positive and one negative ...

**33**

votes

**4**answers

936 views

### If $A,B$ are upper triangular matrices such that $AX=XA\implies BX=XB$ for upper triangular $X$, is $B$ a polynomial in $A$?

A professor of mine told me that this is true, but he doesn't remember what the proof was or where to find it, and I haven't been able to find a source for it yet. As such I am looking for one here.
...

**3**

votes

**1**answer

97 views

### 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{...

**17**

votes

**2**answers

1k views

### How to prove positivity of determinant for these matrices?

Let $g(x) = e^x + e^{-x}$. For $x_1 < x_2 < \dots < x_n$ and $b_1 < b_2 < \dots < b_n$, I'd like to show that the determinant of the following matrix is positive, regardless of $n$:
...

**7**

votes

**0**answers

171 views

### A conjecture on simplex

Let $A_0A_1...A_n$ be a simplex in $\Bbb E^n.$ Let $B_{ij}$ be a point on the edge $A_iA_j,\ 0\le i\not=j\le n.$
Denote by $\beta_i$ the hyperplane passing through the points $B_{i0},$ $B_{i1},$ $B_{...

**2**

votes

**0**answers

70 views

### Characterizing a subclass of row-orthogonal matrices

Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...

**8**

votes

**1**answer

265 views

### An inequality related to Riesz–Thorin theorem, determinants and $L_p$ norm

Let $a, b, c \in \mathbb{R}^n$ , $p \in [1, +\infty)$, prove that
$$\left( \sum_{1\leq i < j <k \leq n} \left| \det\left(\begin{matrix} a_i & b_i & c_i \\
a_j & b_j & c_j \\
...

**0**

votes

**1**answer

103 views

### Stability of a matrix product

Motivation: I am working on a research problem and have been stuck for a while. I hope someone can help, as it requires only linear algebra. :)
Let $H$ be a real, invertible and positive semi-...

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votes

**0**answers

45 views

### Relations among hyperplane mirror symmetries

Let $H(n) \subset O(n)$ be the set of mirror symmetries in $\mathbb{R}^n$ with respect to $(n - 1)$-planes containing the origin. One can see that for any $a, b, c \in H(2)$ we have $abcabc = id$, ...

**4**

votes

**1**answer

137 views

### Positive definite matrices diagonalised by orthogonal matrices that are also involutions

Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$.
I want to know when this matrix is also an involution, i.e., $P^2 = I$.
If there is any ...

**2**

votes

**0**answers

74 views

### A system of homogeneous linear equations

This is the "real-life" (but slightly more technical) version of a question I have asked recently.
For a prime $p>10$, let $\mathcal L_X$, $\mathcal L_Y$, and $\mathcal L_Z$ denote the pencils of ...

**4**

votes

**1**answer

155 views

### Uniqueness of diagonalizing a matrix over $\mathbb{Z}_{p^k}$

We know from linear algebra that if an $n \times n$ matrix $A$ over a field $k$ is diagonalizable (that is, there exists $P \in GL_n(k)$ such that $PAP^{-1}$ is a diagonal matrix), then this diagonal ...

**1**

vote

**1**answer

94 views

### Equivalence of matrices over a vector space

I'm trying to characterize equivalence classes of matrices over a vector space.
Specifically, let $V$ be a vector space over a field $K$, let $M \in M_n(V)$ be an $n \times n$ matrix with entries in $...

**6**

votes

**1**answer

191 views

### Branching from $E(6)$ to $SO(10) \times U(1)$

In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...

**-1**

votes

**1**answer

72 views

### Bound for psd-matrix weighted norm of two related vectors

Let two vectors, $\mathbf{x}, \mathbf{y}$ be related as: $0 \leq x_i \leq \lambda y_i$, for some $\lambda > 0$. That is, $\mathbf{x}$ is coodinate-wise dominated by a scaled version of $\mathbf{y}$....

**5**

votes

**1**answer

130 views

### Finding a particular matrix factor

Consider the following Laurent polynomial matrix-valued function in the variable $x\in\mathbb{C}$
$$
A(x) = \begin{bmatrix} 0 & x \\ x^{-1} & 0\end{bmatrix}.
$$
I'm interested in finding a ...

**14**

votes

**2**answers

505 views

### What are the periodic Dyck paths?

I changed the thread completely so that everything is now elementary linear algebra.
A Dyck path of length $n$ is a list of positive integers $[c_1,c_2,...,c_n]$ with $c_i -1 \leq c_{i+1}$ for all $i$...