Questions tagged [linear-algebra]

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

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Maximum number of vectors with bounds on inner products

Suppose there are 2n vectors $\{m_1,m_2,...,m_n\}$ and $\{\mu_1,\mu_2,...,\mu_n\}$. All vectors are in k-dimensional Euclid space $R^k$. The vectors satisfy: $$m_i \mu_j \leq 0, \quad \forall i<j $$...
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Inverse of a matrix defined as a quadratic form

Suppose I have a vector $x$ of $n$ reals, and tensors $A,B$ of respective sizes $(n,k,k,n)$ and $(k,k,n)$, so that both the array-vector product $Bx$ and the quadratic form $x'Ax$ are of size $(k,k)$. ...
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Bound on the trace of inverse matrix

Suppose $A$ is a positive semi-definite matrix and we can bound its trace as $l \le tr(A) \le L$. I am wondering if it is possible to find the upper and lower bounds on the trace of $A^{-1}$ based on $...
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5 votes
2 answers
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Trace identity for $2 \times 2$ reflections

Let $A, B, C \in \mathrm{GL}(2,\mathbb{C})$ be reflections (i.e., their eigenvalues are $\pm 1$). Please show that $$ \DeclareMathOperator\Tr{Tr}\{\Tr(AB)\}^2+\{\Tr(BC)\}^2 + \{\Tr(CA)\}^2 - \{\Tr(AB)\...
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1 answer
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Does the following expectation-based inequality hold?

Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...
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7 votes
1 answer
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Construction of skew-Hadamard matrix of order 292

I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction? According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
1 vote
1 answer
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The eigenvalues of the product $WD$ for some particular matrices

Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0 Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is $$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
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1 answer
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If the spectral radius of matrix $A$ is less than $1$, how to construct a positive definite $Q$ such that $Q - A^{H}QA$ is also positive definite?

It is relatively easy to prove that if there exists a positive definite matrix $Q$ such that $Q - A^{H}QA$ is positive definite, where $A^{H}$ means the conjugate transpose of $A$, then the spectral ...
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Is there a geometric meaning to this algebraic observation about the Takagi decomposition?

Let $S$ be a complex symmetric matrix, i.e. $S = S^T$. The Takagi decomposition is the statement that $$S = UDU^T$$ where $U^* = U^{-1}$ and $D$ is a diagonal (nonnegative real) matrix. Here's an ...
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Intersection of stabilizer group orbits and algebraic variety of decomposable forms

I have been trying to prove/come up with counter examples to the following situation, any help would be very much appreciated. Let $\{E_I\}$ be a basis of $\mathbb R^6$, so that any vector $V\in\...
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The typical homography matrix requires 4 corresponding points to be solved. Given camera aperture, rotation, image center, can i reduce points needed? [closed]

The typical homography matrix requires 4 corresponding points to be solved. Given camera aperture, rotation, image center, can i reduce points needed? The homography matrix can be found from an ...
1 vote
1 answer
53 views

Convergent condition of the high-dimensional submatrix of some orthogonal matrix

Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are $$ \mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
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1 answer
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The spectrum of the product $JA$ where $J=I_n\oplus (-I_n)$

Let $A$ be a real symmetric matrix in $M_{2n}(\mathbb{R})$with $A^2=I_{2n}$. Suppose that the Schur decomposition of $A$ is given by $A=\Lambda^t D \Lambda$. Let us consider the following matrix. $$...
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1 vote
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Closed-form solution of a particular linear program

(Note: I asked a similar question at math.stackexchange but the present one is more precise.) I have a linear program of the form: $$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$ $...
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4 votes
1 answer
133 views

largest eigenvalue of the difference between two quadratic forms

Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix. Is it true that $$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}...
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What does mean módulo 2pi? [migrated]

I was reading a paper and it have a equation inside absolute value with a small 2pi on the right corner , the paper explains |.|2pi denotes modulo 2pi
3 votes
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Recovering the matrix when the Schur decomposition of its blocks are known

Let E be a real symmetric matrix in $M_n(\mathbb{R})$ where $ n=2m$ and $$E=\left(\begin{array}{cc} G & X \\ X^t & H \end{array}\right)$$ where $G,H,X$ are $m\times m$ matrices. Suppose that $...
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2 votes
1 answer
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Deviation of random matrix from its expectation informs the positiveness of its second smallest eigenvalue

Let $A$ be a PSD random matrix, which has $0$ as one of its eigenvalues. The second smallest eigenvalue of the expectation of $A$ writes as $\lambda_2(\mathbb{E}(A))>0$. Why the following statement ...
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Testing a condition in linear algebra involving Krylov subspaces

Let $A \in \mathcal{M}_n(\mathbb{R})$ be a real-valued $n \times n$ matrix. For $b \in \mathbb{R}^n$, I consider the Krylov subspace $$K_A(b) = \operatorname{span} \{ b, A b, \dotsc, A^{n-1} b \}.$$ ...
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A property of quadratic optimization solution map

Define for a given $\mathbf{b}$ the quadratic optimization solution map by $$ \mathcal{S} \ \colon \ U \mapsto \mathbf{x} \ \colon \ \mathbf{x}^\top \, U^{-1} \, \mathbf{x} = \min_{\mathbf{y} \geq \...
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Lower bounds on eigenvalues of sum of two matrices (one of them is symmetric)

Let $L$ be a matrix with eigenvalues($\lambda$ $\geq$ 0). If I add a constant value (say $a$) to all the elements of $L$, what can we say about the minimum eigenvalue of this perturbed matrix? Note: $...
2 votes
1 answer
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Generate a low-rank sparse covariance matrix

May I ask how to generate a low-rank sparse covariance matrix? Thank you!
2 votes
0 answers
99 views

How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
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Property of positive semi-definite

Let $A$ is a positive semi-definite matrix like this: $$ A = \begin{bmatrix} 1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\ \alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\ \...
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0 answers
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Reference for (general case) of uniqueness of singular value decomposition (SVD)

My statistics research requires me to understand the non-uniqueness of SVD in the degenerate case of repeated singular values. I believe that the statements and proofs on this StackExchange posts are ...
4 votes
1 answer
184 views

Action of complex torus on a vector space

Consider a torus $T$ over $\mathbb{C}$. Let $\rho: T\rightarrow \operatorname{GL}_{n}(\mathbb C)$ be a finite dimensional complex representation. Is there an elementary way (undergrad level) to see ...
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3 votes
1 answer
109 views

On the bounds of the sum of the squares of spectral variation of two real symmetric matrices

Suppose $A$ and $B$ are two symmetric real $\{0,1,-1\}$ matrices of order $n$ with diagonal elements as zeros (therefore the traces are zeros) and eigenvalues $\lambda_1\ge \lambda_2\ge \dotsb \ge \...
5 votes
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285 views

Decomposition of a determinant

Let $M$ be a $4\times 4$ symmetric matrix whose entries $m_{i,j}$ for $i,j =1,\dots,4$ are homogeneous polynomials of degree $2$ in $3$ variables. Assume that $m_{1,1} = 0$. Does there exist a ...
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3 votes
1 answer
174 views

The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$

Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
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1 vote
2 answers
294 views

Condition for equality of modules generated by columns of matrices

Let $R$ be a commutative ring with unit. Let $M_A$ denote the submodule of $R^m$ generated by columns of a matrix $A$ with entries in $R$. Suppose we are given two matrices $A,B \in R^{m \times k}$. I ...
5 votes
1 answer
131 views

Questions about hermitian positive semidefinite matrices

Motivation: I am faced with a $5 \times 5$ hermitian positive semidefinite matrix, depending on parameters, and I wish to show that it is positive definite, for any points in the parameter space (I ...
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2 votes
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A generalisation of Moore-Ore criterion?

Let $K$ be a field of characteristic $p>0$ and $q=p^f$. One supposes that $\mathbb F_q$ is embedded in $K$. One considers elements of $K$: $w_1,\dotsc,w_s$. Assume there exists $b_1,\dotsc,b_s$ in $...
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2 votes
1 answer
237 views

The eigenvalues of the matrix $\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$

What are the eigenvalues/eigenvectors of the matrix $A=\Big(\frac{1}{\cos(k-l)\frac{\pi}{n}}\Big)_{k,l=1}^n$ when $n$ is odd?
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3 votes
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Can the Jordan decomposition of a matrix be computed in a backwards stable way?

Let $PJP^{-1}$ denote the Jordan decomposition of $M$. The matrix $J$ is a direct sum of Jordan blocks; it is unique up to permutation of the Jordan blocks. The matrix $P$ is not unique. There are two ...
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1 vote
1 answer
110 views

The dimension of the eigenvector space of non-negative irreducible matrices

Let $A$ be an irreducible non-negative matrix. Is it true that the eigenvectors of $A$ can span the $R^n$ ? Or are all the eigenvalues of $A$ distinct?
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1 answer
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A question on linear algebra over non-Archimedean local field

Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it ...
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1 answer
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Is there a redundant constraint in linear programming? [closed]

From wikipedia: But... Why do we need the $x\ge 0$ part? We can instead do $-x\le 0$, and thus saving a line in the definition (which is not a big deal but nevertheless nice). (In order to do that, ...
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0 answers
39 views

What information concerning the eigen-structure are transformed on the antidiagonal submatrices?

Let us fix symmetric matrices $A_1$ $A_2$ in $M_m(\mathbb{R})$ with $A^2_1=\alpha I$ and $A^2_2=\beta I$ for some positive $\alpha ,\beta$. For a given matrix $B\in M_m(\mathbb{R})$, let us ...
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Positive linear recurrent sequence

Suppose there is a linearly recurrent sequence $a_k$ satisfies $a_k\geq 0$ and $\sum_{k=1}^{\infty}a_k=1$. Can we always find a $x$ and $r<1$ such that $a_k\leq x r^k$?
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2 votes
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How to decompose a matrix over a ring $F[X_1,\ldots,X_k]$ as a product of two matrices

Let $F$ be a field. Assume any reasonable conditions if needed, such as $F=\mathbb R$, $F=\mathbb C$, $F$ is a finite field, or $F$ has a specific characteristic, etc. Let $C$ be an $n\times1$ matrix ...
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What are the properties of square-matrix algebra with this equivalence class?

Consider the set of all square matrices with the following equivalence class: $\mathbf{A}\sim\mathbf{A}\otimes\mathbf{I}_n$ (or, alternatively, as user @M.G. proposed, $\mathbf{A}\sim\mathbf{I}_n\...
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1 vote
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Solve linear matrix equation involving convolution

I am facing following equation: $$ A * X + C \cdot X = D $$ with: $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure, $X \in \mathbb{R}^{n \times n}$ the ...
1 vote
2 answers
81 views

Transforming matrix to off-diagonal form

I wonder if one can write the following matrix in the form $A = \begin{pmatrix} 0 & B \\ B^* & 0 \end{pmatrix}.$ The matrix I have is of the form $$ C = \begin{pmatrix} 0 & a & b & ...
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2 votes
1 answer
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What is the "Toeplitz matrices from the discrete gradient operators with forward difference"

Recently I have read many papers which focusing on the image enhancement. The description, $D$ are the Toeplitz matrices from the discrete gradient with forward difference occurs many times. An ...
8 votes
2 answers
367 views

Use random inner product to test if at least one vector is uniform

Let $u$ and $v$ be two vectors in $\mathbb{C}^n$. Define a permutation of a vector $v':=\sigma(v)$ by $v'_j = v_{\sigma(j)}$ for any $\sigma \in S_n$. It is easy to show the following for $u,v \in \...
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1 vote
1 answer
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Some calculations with polynomials in the proof of the Routh-Hurwitz test

In an article on the Routh-Hurwitz test, I couldn't see why the following result is true: Let $$p(s)=a_ns^n+a_{n-1}s^{n-1}+\dots+a_{1}s+a_0$$ and $$ \eta_{*} = \frac{a_n}{a_{n-1}}$$ $$g_{\eta}(s):=p(...
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0 answers
93 views

Finding the eigenvectors of a submatrix

Let $A=(a_{kl})$ be a matrix in $M_n(\mathbb{R})$ when $n$ is even. Let $B=(b_{kl})$ be the symmetric $2n$ by $2n$ matrix whose entries are given by, $b_{k,l}=a_{kl}$ if $1\leq k,l\leq n$. $b_{n+k,l}=...
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1 answer
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Tensor nuclear norm for a binary 3rd-order tensor

I am interested in the low-rank approximation of a binary(01) third-order tensor. Does anyone know how to define its tensor nuclear norm based on whatever tensor decomposition methods? Could anyone ...
4 votes
0 answers
133 views

Algorithm for generalized Hilbert's Theorem 90 over $\Bbb R$

$\newcommand{\GL}{\operatorname{GL}} \newcommand{\R}{{\Bbb R}} \newcommand{\C}{{\Bbb C}} $For a natural number $n$, let $z\in \GL(n,\C)$ be a 1-cocycle of $G=\GL_{n,\R}\,$, that is, an invertible ...
0 votes
1 answer
71 views

Can this function be simplified to use only quadratic, linear terms of M with given conditions?

Given the function $$ E(M) = \sum_{i=1}^N \sum_{a=1}^K \left( M_{ia} \cdot \left\lVert\sum_{i=1}^N M_{ia}\cdot x_i\right\rVert_2^2 \right) $$ $x$ is a given constant matrix, $x_i$ is a the $n_\text{th}...

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