Questions tagged [linear-algebra]
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
5,296
questions
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14
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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:
$$m_i \mu_j \leq 0, \quad \forall i<j $$...
- 1
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votes
0
answers
34
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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 $(k,k)$.
...
- 641
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1
answer
74
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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 $...
- 101
5
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2
answers
330
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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
$$ \DeclareMathOperator\Tr{Tr}\{\Tr(AB)\}^2+\{\Tr(BC)\}^2 + \{\Tr(CA)\}^2 - \{\Tr(AB)\...
- 59
1
vote
1
answer
67
views
Does the following expectation-based inequality hold?
Let $\mathcal{F}$ be the space of all functions that uniformly and independently map the alphabet $\mathcal{X}$ to the set $\{1,2,\ldots,A\}$. Let $p(x|y)$ be an arbitrary conditional probability ...
- 63
7
votes
1
answer
221
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Construction of skew-Hadamard matrix of order 292
I am currently looking into how to construct a skew-Hadamard matrix of order 292. Where can I find such construction?
According to multiple papers (e.g. Koukouvinos and Stylianou - On skew-Hadamard ...
- 101
1
vote
1
answer
56
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The eigenvalues of the product $WD$ for some particular matrices
Let $D$ be a diagonal matrix in $M_{2n}(\mathbb{R})$ such that $D^2=I$ and Trace$(D)$=0
Suppose that $e_k$s are the standard vectors in $\mathbb{R}^{2n}$, that is
$$e_k=(0,\cdots 0,1,0,\cdots,0)^t$$...
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5
votes
1
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100
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If the spectral radius of matrix $A$ is less than $1$, how to construct a positive definite $Q$ such that $Q - A^{H}QA$ is also positive definite?
It is relatively easy to prove that if there exists a positive definite matrix $Q$ such that $Q - A^{H}QA$ is positive definite, where $A^{H}$ means the conjugate transpose of $A$, then the spectral ...
- 53
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26
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Is there a geometric meaning to this algebraic observation about the Takagi decomposition?
Let $S$ be a complex symmetric matrix, i.e. $S = S^T$. The Takagi decomposition is the statement that
$$S = UDU^T$$
where $U^* = U^{-1}$ and $D$ is a diagonal (nonnegative real) matrix.
Here's an ...
- 4,058
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0
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60
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Intersection of stabilizer group orbits and algebraic variety of decomposable forms
I have been trying to prove/come up with counter examples to the following situation, any help would be very much appreciated.
Let $\{E_I\}$ be a basis of $\mathbb R^6$, so that any vector $V\in\...
-4
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0
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28
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The typical homography matrix requires 4 corresponding points to be solved. Given camera aperture, rotation, image center, can i reduce points needed? [closed]
The typical homography matrix requires 4 corresponding points to be solved. Given camera aperture, rotation, image center, can i reduce points needed?
The homography matrix can be found from an ...
1
vote
1
answer
53
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Convergent condition of the high-dimensional submatrix of some orthogonal matrix
Let $\mathbf{V}$ be a $p\times p$ orthogonal matrix (i.e., $\mathbf{V}\mathbf{V}^\top = \mathbf{V}^\top \mathbf{V} = \mathbf{I}$) whose columns are
$$
\mathbf{V} = \begin{bmatrix} \mathbf{v}_1 & \...
- 262
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votes
1
answer
63
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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.
$$...
- 3,583
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112
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Closed-form solution of a particular linear program
(Note: I asked a similar question at math.stackexchange but the present one is more precise.)
I have a linear program of the form:
$$\text{minimize} \space\space x_1 \space\space \text{subject to:}$$
$...
- 402
4
votes
1
answer
133
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largest eigenvalue of the difference between two quadratic forms
Let $U,V\in\mathbb{R}^{4\times n}$ such that $UU^T=VV^T=I$, and $A\in\mathbb{R}^{n\times n}$ be an Hermitian matrix.
Is it true that
$$\sqrt{\lambda_{\text{max}}\left(\left(UAU^T-VAV^T\right)^2\right)}...
- 505
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0
answers
13
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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
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86
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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 $...
- 3,583
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votes
1
answer
56
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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 ...
- 43
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28
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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
$$K_A(b) = \operatorname{span} \{ b, A b, \dotsc, A^{n-1} b \}.$$
...
- 559
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0
answers
93
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A property of quadratic optimization solution map
Define for a given $\mathbf{b}$ the quadratic optimization solution map by
$$
\mathcal{S} \ \colon \ U \mapsto \mathbf{x} \ \colon \ \mathbf{x}^\top \, U^{-1} \, \mathbf{x} = \min_{\mathbf{y} \geq \...
- 99
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votes
1
answer
66
views
Lower bounds on eigenvalues of sum of two matrices (one of them is symmetric)
Let $L$ be a matrix with eigenvalues($\lambda$ $\geq$ 0). If I add a constant value (say $a$) to all the elements of $L$, what can we say about the minimum eigenvalue of this perturbed matrix?
Note: $...
2
votes
1
answer
52
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Generate a low-rank sparse covariance matrix
May I ask how to generate a low-rank sparse covariance matrix? Thank you!
- 21
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answers
99
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How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]
I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
- 183
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answers
113
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Property of positive semi-definite
Let $A$ is a positive semi-definite matrix like this:
$$ A = \begin{bmatrix}
1 & \alpha_{1,2} & \alpha_{1,3} & \alpha_{1,4}\\
\alpha_{1,2} & 1 & \alpha_{2,3} & \alpha_{2,4}\\
\...
- 13
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answers
47
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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 ...
- 3,223
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 \...
- 61
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285
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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 ...
- 433
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votes
1
answer
174
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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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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 ...
- 317
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votes
1
answer
131
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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 ...
- 3,959
2
votes
0
answers
35
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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 $...
- 3,167
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votes
1
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237
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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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110
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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 ...
- 4,058
1
vote
1
answer
110
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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?
- 115
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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
85
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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, ...
0
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0
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39
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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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66
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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
$a_k\leq x r^k$?
- 1,405
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answers
70
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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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68
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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\...
- 8,682
1
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0
answers
42
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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 ...
- 11
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
$$ C = \begin{pmatrix} 0 & a & b & ...
- 466
2
votes
1
answer
32
views
What is the "Toeplitz matrices from the discrete gradient operators with forward difference"
Recently I have read many papers which focusing on the image enhancement. The description,
$D$ are the Toeplitz matrices from the discrete gradient with forward difference
occurs many times.
An ...
- 21
8
votes
2
answers
367
views
Use random inner product to test if at least one vector is uniform
Let $u$ and $v$ be two vectors in $\mathbb{C}^n$.
Define a permutation of a vector $v':=\sigma(v)$ by $v'_j = v_{\sigma(j)}$ for any $\sigma \in S_n$.
It is easy to show the following for $u,v \in \...
- 1,541
1
vote
1
answer
73
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Some calculations with polynomials in the proof of the Routh-Hurwitz test
In an article on the Routh-Hurwitz test, I couldn't see why the following result is true:
Let $$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(...
- 11
0
votes
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}=...
- 3,583
0
votes
1
answer
48
views
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 ...
- 12.4k
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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}...
