All Questions
5,882 questions
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205
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Prove that sum of eigenvalues of the inverse of an nxn correlation matrix A is greater than or equal to n
I stuck on this question and here is my thoughts:
So we have a nxn correlation matrix A with eigenvalues: λ_1,λ_2,...,λ_n
1.According to the property of correlation matrix, (λ_1)+(λ_2) + ... + (λ_n) = ...
- 1
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votes
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answers
44
views
Sufficient conditions to order the solutions to a system of linear equations
A pretty elementary question, but does anyone know of sufficient conditions to order the solutions of a system of linear equations? For example, in the system, \begin{align*}\begin{bmatrix}a_{11}&...
- 1
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votes
0
answers
172
views
A characterisation of full subgroups of $\mathrm{GL}_n(\mathbf{F}_p)$
Let $p\geq 5$ be a prime and $\mathbf{F}_p$ a finite field of characteristic $p$. A subgroup of ${\rm GL}_n(\mathbf{F}_p)$ is full if it contains ${\rm{SL}}_n(\mathbf{F}_p)$. When $n=2$, we have the ...
- 667
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0
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107
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The eigenstructure of the symmetric tridiagonal matrix whose entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and $$a_{1,2}=\cdots=a_{n-1,n}=1$$
Suppose that $A=(a_{kl})_{k,l=1}^n$ is a symmetric tri-diagonal matix in $M_n(\mathbb{R})$ whose diagonal entries are $a_{kk}=\cos\frac{k\pi}{n+1}$ and
$$a_{1,2}=\cdots=a_{n-1,n}=1$$
Any approach to ...
- 4,058
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0
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124
views
Eigenvectors of the symmetric tridiagonal matrices whose entries above the diagonal are all the same
Let us consider the real symmetric tridiagonal matrix $T=(t_{kl})$ in $M_n(\mathbb{R})$ with
$$t_{1,2}=t_{2,3}=\cdots=t_{n-1,n}=1$$
How can we compute the eigenvectors of $T$?
- 4,058
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0
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126
views
On the absolute difference of the Laplacian eigenvalues of an unbalanced signed graph and its underlying graph
Let $\Sigma=(G,\sigma)$ be an unbalanced signed graph with the underlying connected graph $G=(V,E)$ and $\sigma:E\rightarrow \{-1,1\}$, the signing function. Let the Laplacian eigenvalues of $\Sigma$ ...
- 61
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answers
117
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Covering zeroes of quadratic forms by linear forms
Consider a quadratic form
$$Q(x_1,\ldots,x_n)=\sum_{i,j}a_{ij}x_ix_j,$$
where $a_{i,j}\in \mathbb{R}$ and $x_i\in A$ for some $A\subset \mathbb{R}$ such what $|A|=k$.
Question. What is the smallest ...
- 2,288
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145
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Bound on solutions of $Ax \ge b$
Let $A \in \mathbb{Z}^{m \times n}, b \in \mathbb{Z}^{m \times 1}$.
One can show that if there is a solution of $Ax \ge b, x \in \mathbb{R}^n$ then there is one such that $\|x\|_{\infty} \le c (\|A\|_{...
- 439
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0
answers
112
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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\...
- 10.1k
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273
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}=...
- 4,058
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votes
1
answer
70
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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 ...
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1
answer
80
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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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votes
1
answer
104
views
Do subgradient inequalities hold for matrix convex functions?
Suppose $f$ is a matrix convex function over symmetric, positive semidefinite matrices with spectra in some interval $I$ [1]. That is, for $A,B\succeq 0$ with spectra in $I$, and any $\theta\in[0,1]$,
...
- 253
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149
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Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
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0
answers
232
views
How to analyse the range of eigenvalues of a symmetric and indefinite matrix?
Let $G$ be a symmetric and indefinite matrix defined as follows
$$ G := S - \begin{pmatrix} I_n & I_n \\ I_n & I_n \end{pmatrix},$$
where $S$ is a symmetric positive definite matrix of size $...
- 1
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1
answer
74
views
$\det(HH’) = 0$ for nonnegative $H$
$H$ is an $n\times m$ matrix with non-negative coefficients and $n < m$. $H'$ is the transpose of $H$.
Are the following statements true?
If $\det(HH’) > 0$, the rows of $H$ define the edges of ...
- 13
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answers
78
views
Topology of independence set of a vector space
This seems like something that would have a well-known treatment somewhere, but I'm not sure where to look. If we have a vector space $V$ (or maybe even a module), we can consider an abstract ...
- 2,044
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104
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Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$
Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define
$$
u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).
$$
Question. What are necessary and ...
- 6,853
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174
views
Lipschitz map on positive definite cone of $n$-by-$n$ matrices
A function matrix $f : X \to \mathbb R$ is a convex Lipschitz continuous matrix function with Lipschitz constant $\mathrm L$ with respect to a fixed given norm $\|\cdot\|$, i.e., $|f(A)-f(B)| \leq \...
- 91
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168
views
Multiplying by Loewner-ordered matrices
Suppose $A$ and $B$ are symmetric positive (semi-)definite, and $A<B$ in Loewner order, meaning $B-A$ is positive (semi-)definite. Is it true that, for a symmetric positive-definite $C$, we have $...
- 93
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108
views
Solving a nonlinear equation maybe with Lambert W function
Can you please help me solve the following nonlinear equation?
\begin{equation}
\boldsymbol{z} \odot\left(\boldsymbol{\Gamma}^{\top} \boldsymbol{y}\right)=(\beta)^{\frac{1}{m-1}}\left(\frac{m-1}{...
- 11
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answers
144
views
Given a binary parity check matrix, find a parity check matrix for the same code such that no row has weight greater than $k$
A parity check matrix for a binary linear code is a matrix $H$ (with linearly independent rows) such that the (right) kernel of $H$ is the codespace. Clearly we can replace any row $r_i$ by $r_i + \...
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156
views
Optimal solution of complex optimization problem
Let $Q(x)=a(x)e^{jb(x)}$ be a complex function of $x$. We want to approximate this function with $R(x)=\alpha e^{jx\beta}$ such that
\begin{align}
\text{arg}\min_{\alpha,\beta} \int_{-\frac{A}{2}}^{\...
- 287
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answers
114
views
Degeneracies in linear combination of tensor product of Pauli matrices
Let $P_i \in \{I,X,Y,Z\}^{\otimes n} $, that is $P_i = \bigotimes_{i =1 }^n \sigma_i$ with $\sigma_i \in \{I,X,Y,Z\}$, where
$$
I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \hspace{1cm} X =...
- 131
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0
answers
37
views
Linear system with +-1 coefficients and three variables for each equation
I have a linear system $LS$, where each equation contains three variables and the coefficient of each variable is $\pm 1$. For example, I have $x_{a}-x_{b}+x_{c}=p$ ($p$ is the known term).
Suppose ...
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answers
99
views
Invertible matrices with bounded nonnegative coefficients
I am teaching a class in linear algebra and I asked myself the following question: what is the chance to get an invertible matrix if I write a random one? My impulsive answer is "very likely"...
- 2,152
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92
views
Positive definite matrix and Hörmander theory
Let $\varphi \in C_{0}^{\infty}, \varphi\neq 0$. We'll consider the inner product in $L^{2}.$
Let $\alpha,\beta$ multi-index, $m\in \mathbb{N}$ such that $|\alpha|,|\beta|\leq m$ and set
$$
\varphi_{\...
- 101
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56
views
Determining the total number of nonzero expansion terms in a (0,1)-matrix
Let $A=(a_{ij})_{n\times n}$ be a $(0,1)$-matrix such that it contains equal number of $1$s in each row and column. Is there any general method to count the total number of the nonzero terms $\prod_{i=...
- 133
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151
views
CRT for linear forms
Suppose $A,B,C,A',B',C'$ are random distinct primes in $[T,2T]$ and $u,v$ are integers in $[T,2T]$.
Suppose we know:
$$Au+Bv\equiv r\bmod C$$
$$A'u+B'v\equiv r'\bmod C'$$
can we identify $u,v$ in ...
- 13.9k
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180
views
Adding the AWGN to the data makes its covariance matrix always positive definite?
I'm working on a numerical method that estimates direction-of-arrivals in antenna arrays.
I realized that every time I add the AWGN (Additive white Gaussian noise) to a data (which is a matrix), its (...
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0
answers
92
views
Classification of elements $GL(d, \mathbb{R})$
Any $SL(2, \mathbb{R})$ is either elliptic or hyperbolic, or parabolic up to conjugacy; see here.
Do we have the same classification for $GL(d, \mathbb{R})$? If so, could you please introduce some ...
- 1,043
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0
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57
views
Determinant of barycenter of a hyperbolic-matrix
Let $A \in \operatorname{GL}(d, \mathbb{R})$ be a hyperbolic matrix. I want to show that $$\det((1-\alpha)A+\alpha\operatorname{Id})\geq 1,$$
where $0<\alpha<1$.
Attempt:
In $\operatorname{SL}(2,...
- 1,043
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0
answers
67
views
Integration of matrix form of Vasicek variance (Python/Matlab)
$X_t$ is a vector and follows the following Vasicek process.
$$
dX_t=(mu-K\cdot X_t)dt+Sigma_x\cdot dZ_t \\
$$
What is the variance of $X_t$?
In scalar form the answer is $\frac{Sigma_x^2}{2\cdot K}\...
- 1
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1
answer
397
views
What is the best way to choose initial basis when applying simplex method to an equality form of LP?
Currently I'm trying to write a practically fast LP solver for a sparse instance, which is by simplex method with LU decomposition and eta-matrix update. In the development I realized that I'm not ...
- 1
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0
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133
views
A question from Richard Hamilton's paper "A matrix Harnack estimate for the heat equation"
Richard Hamilton "A matrix Harnack estimate for the heat equation. Communications in Analysis and Geometry. 1(1993), 113-126."
On page 125, at the end of the proof of Theorem 4.3, I abstract ...
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0
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105
views
Unitarily equivalent matrices that are also unitarily equivalent on orthogonal subspaces
Consider two positive semidefinite matrices $A$ and $B$ on $\mathbb C^d$.
Let $\{P_i\}_{i=1}^m$ be a complete family of $m$ orthogonal projectors on $\mathbb C^d$ (i.e., $P_i^*=P_i, P_iP_j=\delta_{ij}...
- 31
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124
views
The best unitary matrices that approximate a matrix product
Let $\mathbf{A}$ be an arbitrary $N\times N$ complex matrix. Moreover, $\mathcal{U}_1$ and $\mathcal{U}_2$ are distinct subsets of all unitary matrices. Suppose the matrices $\mathbf{U}_1$ and $\...
- 287
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46
views
What kind of bounds for $\mathrm{Re}(\lambda(A))$ when $\lambda_{\mathrm{max}}(A + A^t) < 0$?
What can be said about the real parts of eigenvalues of $A \in \mathbb R^{n\times n}$ when $\lambda_{\max}(A + A^t) < 0$?
I think the real parts of eigenvalues of $A$ will be negative, but I can't ...
- 1
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0
answers
80
views
Multivariate vandermonde matrix [duplicate]
I am dealing with a generalized vandermonde matrix, and i wandered if there were simple results availiable on it's invertibility.
Fix a dimension $d$, and let $\mathbf x_1,...\mathbf x_n$ be points in ...
- 686
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112
views
How to prove or disprove $ |r'(a)| \leq c_1 \sup_{x \in [a-1/2,a+1/2]} |r(x)|$?
Let $ A = \begin{bmatrix}
a & 1 \\ 0 & a
\end{bmatrix}$ be a Jordan matrix with $ -1 < a < 1 $.
Let $r(z) = \frac{p(z)}{q(z)}$ be an irreducible rational function, where $p(...
- 101
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0
answers
248
views
Is there a relationship between infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I wish to know if there is a known relationship between the infinity norm (or any other norms) of a vector and the trace of its covariance matrix?
I have found a paper that used the following ...
0
votes
0
answers
112
views
Existence of a subspace of having no isotropic 2-plane
Let $V$ be a vector space of dimension $n$ over the field $\mathbb {Q} $. A subspace $W$ is isotropic for a skew-bilinear form $\alpha$ on $V$ if $\alpha(x,y) = 0$ for all $x,y \in W$.
More ...
- 923
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0
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198
views
eigenvalues of the product of a unitary with a diagonal
In $M_n(\mathbb{C})$, suppose $U$ and $D$ are a unitary and an invertible diagonal matrix with eigenvalues $\{e^{i\theta_1},\cdots,e^{i\theta_n}\}$ and $\{e^{i\eta_1},\cdots,e^{i\eta_n}\}$ ...
- 4,058
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0
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253
views
Determinant of chain complexes
Let $\mathcal{C}$ be the category of bounded cochain complexes of $R$-modules for a commutative ring $R$. I am trying to prove the following formula involving determinant $\text{Det}(F)$ of a map of ...
user30211
0
votes
0
answers
44
views
Polynomial representation with shared root
Let $R$ be a polynomial quotient ring of the form $F_3[x_1,x_2,...,x_n]/\langle \{x_i^3-x_i\}_{1\le i\le n} \rangle$ and $\{f_i\}_{1\le i \le m}$ be elements of $R$, where $m > n$. We know that the ...
0
votes
0
answers
150
views
Sequence in matrices converging to Identity
Let $M_{k}(\mathbb{C})$ be the set of $k \times k$ complex matrices. I am trying to find a sequence of polynomials $P_{n}: M_{k}(\mathbb{C}) \to M_{k}(\mathbb{C})$ (or continuous functions $f_{n}$) ...
- 31
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0
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96
views
Books on limiting properties of matrices with growing size
This question has been posted on Math-Se previously.
I am studying asymptotic properties of the Projection Matrix
$$
H_n=X'(X'X)^{-1}X
$$
By the Gerschgorin disc theorem, the bounds on the ...
- 101
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votes
0
answers
41
views
Combining coefficients in a sum of inequalities
In an induction proof of a lemma I would like to prove the following statement.
Let $U$ be a non-empty finite set and let $X_{i,j}$ for all $i\not=j \in U$ be real numbers. Assume for each $j\in U$ we ...
0
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0
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257
views
What's the definition of Euclidean density?
In page 19 of the article "Equivariant cohomology with generalized coefficients" the authors say:
Let $V $ be an n-dimensional real vector space, let $v'$ be a non-zero element in $ \wedge^...
- 139
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0
answers
54
views
Rank decomposition of matrices over $\mathbb F_2$
Given an integer matrix $M\in\mathbb Z^{n\times n}$ of real rank $k$ what is the minimum and maximum number of rank $1$ matrices $B_1$ to $B_t$ we require so that $M\equiv\sum_{i=1}^tB_i\bmod 2$?
If $...
- 13.9k