All Questions
608 questions
3
votes
1
answer
712
views
Relation between eigenvalues and the gram matrix
We have matrix data $X$ which is $n\times d$. We use the covariance matrix/ design matrix/ gram matrix $X^T X$ to perform least-squares/ PCA. I compute the eigen basis representation of said matrix
$$...
- 215
3
votes
0
answers
384
views
Systems of linear octonionic equations
Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a ...
- 75
3
votes
0
answers
130
views
The probability that the dominant eigenvalue of a random real matrix is real
Let $X_n$ be an $n\times n$ real matrix where the entries in $X_n$ are independent, normally distributed, have mean $0$, and variance $1$. Suppose that $\lambda_1,\dots,\lambda_n$ are the eigenvalues ...
3
votes
1
answer
272
views
Enumerating possible number of satisfied linear equations
Consider a system of linear equations of variable $x=(x_1,\cdots,x_n)$ where each $x_i\in\{ 0,1,\cdots,L-1 \}$. Clearly, there are $\frac{n(n-1)}{2}$ number of equations in the system.
$$x_i-x_j=0, \ \...
- 405
3
votes
0
answers
263
views
Does the product of principal sines between subspaces satisfy the triangle inequalilty?
As we know that the volume of a matrix $X$ is defined as $\sqrt{\det(X^TX)}$, if we consider the volume of two matrices $X$ and $Y$, with $X,Y \in \mathbb{R}^{N\times d},d<N,\dim(X)=\dim(Y)=d$, ...
- 131
3
votes
1
answer
248
views
Log-convexity of conditional variances
Let $K$ be a positive integer and $C$ be any $K \times K$ non-singular matrix. For positive real numbers $q_1, \dots, q_K$, define
$$\Sigma(q_1, \dots, q_K) = CC' + diag(\frac{1}{q_1}, \dots, \frac{1}...
- 765
3
votes
2
answers
2k
views
Invariants of Matrix Reordering
are there any invariants of matrices, that are not affected by row- and/or column permutations?
To me it seems that the sequence of singular values could be such an invariant; am I right, resp. are ...
- 13.2k
3
votes
1
answer
333
views
Maximising a Rayleigh quotient over a subspace
Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...
- 215
3
votes
1
answer
655
views
Upper bounds on the condition number of the eigenvector matrix
Let $A$ be an $n\times n$ real matrix with entries in a fixed interval $[a_\min,a_\max]$, with $a_\min$, $a_\max>0$.
Question: Are there any upper bounds on the condition number of the ...
- 2,712
3
votes
1
answer
784
views
Expected number of random binary vectors so that the form a basis
I would like to compute the expected number of vectors in $\mathbb{F}_2^n$ we need to draw (following a uniform distribution) so that they form a basis of $\mathbb{F}_2^n$, i.e., that we have $n$ ...
- 43
3
votes
2
answers
412
views
Indecomposable integral representations of a group of order 2 "by hand"
This question is a duplicate of
that 2010 MO question.
I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$.
Clearly, ...
- 14.2k
3
votes
1
answer
268
views
How to prove that for the real stable characteristic polynomial $P=\Phi_T$ of a tree $T$, $P_iP_j-PP_{ij}=(\Phi_{T-[v_i,v_j]})^2$?
If $G$ is a labeled graph, the multi-affine characteristic polynomial (which depends on labeling) is defined by
$\Phi_G(x_1,...,x_n)=\det(I_x-A)$, where $I_x$ is the diagonal matrix $diag\{x_1,...,x_n\...
- 487
3
votes
1
answer
1k
views
Symplectic block-diagonalization of a complex symmetric matrix
This is a follow-up question to the one asked here:
Given a complex symmetric $2n\times2n$-matrix $A$, i.e., $A\in \mathbb{C}^{2n\times2n}$ with $A = A^T$. Is it possible, to block-diagonalize $A$ ...
- 290
3
votes
0
answers
258
views
Solve $(A+B)x=y$ given Cholesky decomposition of A and B
I wish to solve for $x$ in
$$
(A+B)x=y
$$
given square symmetric matrices $A$ and $B$. For certain reasons I have already computed the Cholesky decompositions for A and B:
$$
A = L^T L
$$
$$
B = M^...
- 561
3
votes
1
answer
1k
views
Number of full-rank binary matrices with no rows of weight 1
For $m > n$, I want to calculate the number of binary matrices with $m$ rows and $n$ columns for which two conditions hold:
the rank of a matrix is $n$ (i.e. it is full-rank, as $m > n$);
there ...
- 367
3
votes
1
answer
2k
views
eigenvalues of product of many symmetric positive definite matrices
Given $A_1, ..., A_n$ ($n\geq 3$), where each $A_i$ is a $d$-by-$d$ symmetric, positive definite matrix, define $S = A_1\cdot A_2\cdot...\cdot A_n$ (product of all the $A_i$'s). Let $\lambda_1(A)$ and ...
- 199
3
votes
3
answers
1k
views
Are the finite dimensional von Neumann algebras, singly generated?
Let $\mathcal{M}$ be a finite dimensional von Neumann algebra, then :
$$\mathcal{M} \simeq \bigoplus_i M_{n_i}(\mathbb{C})$$
Question : Is it singly generated (as von Neumann algebra)? how ?
...
3
votes
1
answer
166
views
The spectral radius of a modified graph
Let $H$ be a graph and let $G=H \vee K_{1}$ be obtained by creating a new vertex and joining it to every vertex in $H$.
This situation has many different names: $G$ is called the cone or the ...
- 6,962
3
votes
0
answers
434
views
Name for matrices with vanishing row and column sums
Question:
is there a special name for matrices whose rows and columns sum to zero?
I actually need information about those matrices and thus a keyword for online search.
Edit:
as there apparently is ...
- 13.2k
3
votes
1
answer
735
views
A similar Cauchy-Schwarz inequality with linear-algebra
Let $A$ be matrix in $M_{n}$ (i.e., $n\times n$ complex matrices), and $\|A\|\le 1$, we call it a contraction.
Assume that $A$ and $B$ are contractions such that
$I-AA^*$ and $I-BB^*$ are positive-...
- 4,280
3
votes
0
answers
237
views
Multi-dimensional permanent of structured tensor
I am facing the multidimensional permanent
\begin{equation} \text{perm}(W) = \sum_{\sigma, \rho \in S_n} \prod_{j=1}^n W_{j, \sigma_j, \rho_j } \end{equation}
of a 3-tensor $W_{j,k,l}$ of ...
- 93
2
votes
1
answer
279
views
Unit singular value conjecture for discrete Fourier transform submatrix
This question was motivated by Singular value decomposition of truncated discrete Fourier transform matrix
Consider for integers $1\leq k\leq N$, $1\leq n_0\leq N-k+1$ the $k\times k$ sub-unitary ...
- 188k
2
votes
0
answers
345
views
Extension of the Gershgorin circle theorem for symmetric matrices and localization of positive eigenvalues
In mathematics, the Gershgorin circle theorem can be used to localize eigenvalues of a matrix (including symmetric). Let $A$ be a real symmetry $n × n$ matrix, with entries $a_{ij}$. For $i∈{1,…,n}$ ...
- 145
2
votes
1
answer
325
views
Determinant and inverse of a "stars and stripes" matrix
This is a variant of another MO question. Consider the matrix
$$M_n:=\begin{bmatrix}c_1& a & b&a& \ddots & a \\ b & c_2 & a& b&\ddots & b\\ a & b & c_3&...
- 13.4k
2
votes
1
answer
133
views
Is the sum of the circulant matrix with a super upper triangular matrix diagonalizable?
By the circulant matrix $C$ in $M_n(\mathbb{R})$, we mean that
$$C=[e_n|e_1|\cdots|e_{n-1}]$$ where $e_1,\cdots,e_n$ are the standard basis vectors in $\mathbb{R}^n$. It is well-known that
$$C=\...
- 4,058
2
votes
0
answers
172
views
Minimum of $\mathrm{rank}\left( \boldsymbol{W} \boldsymbol{H} \right)$, with $\boldsymbol{W}$ block diagonal
Let us assume that we have a full-rank $(n\cdot l)\times k$ matrix, $\boldsymbol{H}$, with no specific structure (e.g., a realization of a Gaussian i.i.d. random matrix), and an $m\times (n\cdot l)$ ...
- 61
2
votes
1
answer
376
views
Maximize inner product of a tensor of unitary matrices
How can one maximize the following function:
$ f(V) = || V \otimes V - U_1 \otimes U_2 ||$
where $U_1, U_2 \in SU(n)$ are given and we seek to maximize over $V \in SU(n)$.
Both the maximum value of ...
- 2,099
2
votes
1
answer
137
views
Design constraint systems over the reals
This question is inspired by the discussion at this problem.
Suppose I have a design consisting of a finite point set $U$ of size $|U|=m_{\emptyset}$ and a family of $n$ subsets (sometimes called ...
- 30.1k
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
2
votes
0
answers
136
views
Linear independence of functions
Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
- 859
2
votes
1
answer
2k
views
How to prove a unit norm matrix is the average of two unitary matrix
How to prove a unit norm matrix is the average of two unitary matrix
- 1,706
2
votes
2
answers
403
views
Is this a linear optimization problem? $Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative
$Ax=0$, $A$ has $m$ rows and $n$ columns, $m \le n$, all entries of $x$ are non-negative.
What should $A$ satisfy to guarantee the equation set have only zero solution?
- 23
2
votes
1
answer
1k
views
Condition for doubly non-negative matrices to be completely positive
Consider a doubly non-negative matrix $A$ of order $n$. $A$ is completely positive if and only if $A$ can be factorized into $BB^{T}$ where all entries in $B$ are non-negative. $B$ is $n\times k$. The ...
- 505
2
votes
0
answers
784
views
Can the matrix exponential be equal to the elementwise exponential [closed]
Just out of curiosity: does there exist a matrix $A=(a_{i,j}) \in \mathbb{C}^{n\times n}, n>1$ such that $(e^{a_{i,j}})\in \mathbb{C}^{n\times n}$ is equal to the matrix exponential $e^A=\sum_{k\...
- 2,286
2
votes
1
answer
143
views
Inequality for hermitian matrices
Let $p_1$ and $p_2$ be a complete system of orthogonal projections on $\mathbf R^n, n \geq 2$ (i.e., $p_i^2=p_i=p_i^*$ and $p_1+p_2=\bf{1}$) and $S_1, S_2$ be two hermitian operators such that $S_i \...
- 73
2
votes
0
answers
96
views
Smallest eigenvalue separation in the Gaussian ensemble of random matrices
This question is motivated by Guido Li's question Expected minimal distance of eigenvalues, which concerns the sum of a deterministic matrix and a Gaussian random matrix. The deformation of the ...
- 188k
2
votes
1
answer
244
views
Expected minimal distance of eigenvalues
Let $A$ be an arbitrary symmetric matrix and $B$ be a random GUE matrix. I would like to know. Are there any results on the minimal eigenvalue distance between two distinct eigenvalues of $A+B$? I ...
- 73
2
votes
0
answers
164
views
Linear combinations of low-degree polynomials with agreement guarantee
Let $\mathbb{F}$ be a finite field of odd characteristic and let $f:\mathbb{F}_{\leq d}[x,y]\rightarrow\mathbb{Z}$ map bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$ to ...
- 125
2
votes
1
answer
665
views
Covering the cone of positive semidefinite matrices by intervals
Is it possible to cover the cone of positive semidefinite matrices by a finite/countable/interesting family of closed intervals of matrices?
How about a general convex cone?
For the finite case the ...
- 6,962
2
votes
1
answer
742
views
rank of a linear combination of matrices
Let $A_1,..., A_s \in M_n(\mathbb{R})$ be symmetric matrices and suppose they are linearly independent over $\mathbb{R}$. This means that
$$
m = \min_{(c_1, ..., c_s) \in \mathbb{R}^s \backslash \{0\}}...
- 3,625
2
votes
1
answer
1k
views
Subgradient of Minimum Eigenvalue
Consider three $N \times N$ Hermitian matrices $A_0$, $A_1$, $A_2$. Consider the function
\begin{align}
f(t_1,t_2)=\lambda_{\text{min}}(A_0+t_1A_1+t_2A_2)
\end{align}
where $\lambda_{\text{min}}$ ...
- 1,421
2
votes
2
answers
235
views
Theoretical/Practical Implications of DFT Eigenvectors
Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
- 4,058
2
votes
1
answer
242
views
Necessary conditions for existence of linear combination of these matrices to be singular
I'm doing some research in Control theory, and a stumbled with this problem. Any help is appreciated.
QUESTION
Let $P_1,\dots,P_m$ be $m$ symmetric positive definite $n\times n$ matrices with $m<n$ ...
- 344
2
votes
1
answer
927
views
Paralel bezier curve
If I have a cubic Bezier curve specified by two endpoints and two control points, how can I find an offset curve which is "parallel" to the original at some given distance, after i have determined the ...
- 123
2
votes
1
answer
385
views
Determinants of striped Hankel matrices
This question is related to the matrices described in Deyi Chen's recent MO post (look at some examples there). The main difference: we are asking for a determinant evaluation instead of a permanent, ...
- 43.2k
2
votes
1
answer
290
views
Endomorphism rings of infinitely generated free modules generated by idempotents?
Let $M$ be a free right $R$-module. When $M_R\cong R_R^n$ with $n\in \mathbb{Z}_{\geq 1}$, then we know that the endomorphism ring $E={\rm End}(M_R)$ is isomorphic to $\mathbb{M}_n(R)$. We also know ...
- 23
2
votes
4
answers
1k
views
Change of variables in a Gaussian integral in matrix form
I have a problem in which I have to compute the following integral: $$\mathop{\idotsint\limits_{\mathbb{R}^k}}_{\sum_{i=1}^k y_i=x} e^{-N^2r(\sum_{i=1}^k y_i^2-\frac{1}{k}x^2)} dy_1\dots dy_k,$$
where ...
- 93
2
votes
2
answers
840
views
When can a matrix with negative entries have a completely non-negative dominant eigenvector?
Perron-Forbenius obviously answers this question for positive and for certain non-negative matrices. I want to know whether these conditions can be weakened at all. In other words, what, if anything, ...
- 31
2
votes
0
answers
78
views
An two-norm estimate for symmetric $k$-tensors
Let $(V, \langle, \rangle)$ be an $n$ dimensional innerproduct space and let $S^k(V)$ denote the space of $k$-fold symmetric tensors. The inner product naturally extends to $S^k(V)$. Denote the ...
- 2,478
2
votes
0
answers
214
views
Is there a symmetric basis for $\mathbf{Q}(x,y)$?
Consider $\mathbf{Q}(x,y)$, the rational functions in $x$ and $y$, as a vector space over $\mathbf{Q}$.
Let $\sigma$ be the map interchanging $x$ and $y$. Is there a basis for $\mathbf{Q}(x,y)$ ...
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