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

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2
votes
1answer
104 views

Number of Matrices with bounded determinant

Here's my question: Let $k,B,C$ be positive integers such that $B<C$. Can you give an upper bound for the number of $k\times k$ integer matrices having entries bounded in modulus by $B$ having ...
-6
votes
0answers
40 views

solve for three unknowns. [on hold]

Apple cost 97 dollars. Orange cost 56 and lemon cost 3. The total amount spent is 16047 dollars and total fruits bought is 240 and each one is bought atleast one. Calculate how many of each have been ...
1
vote
1answer
105 views

Diagonalization of 4th order tensors

I have been wondering about the following problem... Let $n$ be a positive integer and denote by $M_n^s$ the space of symmetric $n\times n$ real matrices. Now, we look at the space $\mathcal ...
0
votes
0answers
222 views

expected matrix inverse of circulant plus diagonal matrix with chi-square variables

Let $R$ be a semi-definite $N\times N$ circulant Toeplitz matrix and let $N\to \infty$. Let $D$ be an $N\times N$ diagonal matrix where the elements on the main diagonal are independent chi-square ...
-1
votes
1answer
40 views

NonLinear Maps and homogeneity [on hold]

An example of a function $\phi : R^2 \to R$ such that $\phi(av) = a \phi(v)$ but $\phi$ is not linear. So I know that I need to find a function that has linear homogeneity but doesn't have the ...
6
votes
2answers
154 views

elementwise functions of positive definite matrix

The fact that the Schur (that is, element wise) product of two positive definite (symmetric) matrices is positive definite immediately implies (using the convexity of the positive semi definite cone) ...
-4
votes
0answers
41 views

let A be an n*n matrix with real entries which of the following is coorect? [on hold]

let A be an n*n matrix with real entries which of the following is coorect? (a) if A^2 =0 then A diagonalisable over complex numbers (b) if A^2= I then A diagonalisable over real numbers (c) if A^2 ...
4
votes
3answers
195 views

Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex \begin{align} \mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\} \end{align} and consider the ...
4
votes
1answer
117 views

Isomorphism of matrix ring over ore domain

Let $R_1,R_2$ be (left and right) ore domains. Does $ Mat_n(R_1)\cong Mat_m(R_2)$ implie m=n and $q.f.(R_1)\cong q.f.(R_2)$? An counter example, a proof or a reference is welcomed. Thanks
-3
votes
0answers
65 views

my question is about matrix [closed]

let A be an n*n matrix with real entries which of the following is correct? 1.if A^2=0, then A is diagonalisable over complex numbers. 2.if A^2=I, then A is diagonalisable over real numbers 3.if ...
7
votes
1answer
123 views

Finite-dimensional inverse limits of double-dual spaces

Let $k$ be a field and $\{V_i\}_{i \in I}$ a filtered projective system of $k$-spaces with transition maps $f_{ji}: V_j \rightarrow V_i$ for $i \leq j$ (for my purposes we may assume the index set is ...
2
votes
2answers
218 views

Finding the set of all $0-1$ vectors in an affine subspace

We are given a $0-1$ matrix $A$ with constant row and column sum, and we need to find out if there exists a $0-1$ vector in the solution space of $Ax = \mathbf{1}$ over $\mathbb{Q}$ (or $\mathbb{Z}$) ...
8
votes
0answers
196 views
+100

Codimension of the range of certain linear operators

Assume that $P(x,y), Q(x,y) \in \mathbb{R}[x,y]$ are two polynomials. We define a linear map $D$ on $\mathbb{R}[x,y]$ with $D(U)=PU_{x}+QU_{y}$. In fact $D$ is the derivational operator correspond ...
8
votes
2answers
698 views

How to transform matrix to this form by unitary transformation?

Without loss of gernerality, we can only consider $n$-dimensional diagonal matrix $M$ whose elements are all nonnegative, i.e. $$M=\operatorname{diag}(m_1,m_2,\cdots,m_n)\ (m_i \geq 0).$$ Then is ...
-4
votes
0answers
43 views

2-norm calculate [closed]

hello all i want to ask if any body know how i can calculate x value in this equation with given y and A ,,, and i have the norm value because first i will estimate the x and then i wan t to calculate ...
2
votes
0answers
26 views

Text book for 2nd Linear Algebra course [migrated]

I stumbled across this site while searching for Hoffman and Kunze. There was a discussion about using HK for a beginning linear algebra course. I am teaching (for the first time) a 2nd course in ...
5
votes
1answer
306 views

Characterizing invertible nonnegative matrices with bounded sums

Almost a year ago, I asked in this question about obtaining a tight bound on the sum of the entries of the inverse of a strictly positive definite matrix. Denis Serre gave a nice counterexample ...
53
votes
10answers
5k views

Geometric proof of the Vandermonde determinant?

The Vandermonde matrix is the $n\times n$ matrix whose $(i,j)$-th component is $x_j^{i-1}$, where the $x_j$ are indeterminates. It is well known that the determinant of this matrix is $$\prod_{1\leq ...
2
votes
1answer
83 views

Comparison of the smallest eigenvalues of two tridiagonal matrices

Let $n\geq2$ be an integer and $E_{ii}$ for an integer $2\leq i\leq n$ be the $n\times n$-matrix with its $ii$-entry equal to 1 and remaining entries equal zero. Furthermore, let ...
0
votes
1answer
91 views

Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation: $X=c \cdot AXA' - diag(c \cdot AXA')+ I$, where (1) $A \in R^{n \times n}$ is a given matrix whose element ...
3
votes
4answers
179 views

Determinant of sum of Kronecker products

Given four real symmetric matrices $A,B \in \mathbb{R}^{n \times n}$ and $C,D \in \mathbb{R}^{m \times m}$, is there an efficient way to compute the determinant: $\det|A \otimes C + B \otimes D |$
0
votes
1answer
62 views

Solution of infinite dimension linear system

Suppose that ${a_n}$ and $b_n$ is decreasing sequence such that $a_0=A$, $lim_{n->\infty}a_n=0$ and $b_0=B$, $lim_{n->\infty}b_n=0$. For fix n, we can construct n dimension linear equation ...
10
votes
3answers
516 views

Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...
5
votes
16answers
2k views

Vector spaces without natural bases

Does anyone know any nice examples of vector spaces without a basis that is in some sense "natural". To clarify what I mean, suppose we look at $\mathbb{R}^2$. We define $\mathbb{R}^2$ as pairs of ...
0
votes
1answer
80 views

Bounded domain matrix factorization

Assume we're trying to find $A\in [-1,1]^{n\times d}, B\in [-1,1]^{m\times d}$, from an observed matrix $C\in [-d,d]^{n\times m}$, where $C=AB^T$. The goal is to return $\widehat A, \widehat B$ such ...
38
votes
2answers
2k views

Geometric interpretation of characteristic polynomial

The coefficients of lowest and next-highest degree of a linear operator's characteristic polynomial are its determinant and trace. These have well-known geometric interpretations. But what about its ...
5
votes
1answer
120 views

Simultaneous Tridiagonalization of a given set of hermitian matrices?

I have a set of $N\times N$ hermitian matrices $A_i,~i=1,\dots,M$. Are there any results on the possibility of simultaneously tridiagonalizing them?
3
votes
1answer
98 views

standard practice for large dense truncated svd computations?

What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices. I assume for k small that you use a ...
0
votes
0answers
18 views

Integer Solutions To Linear Equation [migrated]

$$a*q_1+b*q_2=c$$ $$a*q_3+b*q_4=f$$ $q_1, q_2, q_3, q_4$ rational numbers, $c,f$ integer Given $q_1, q_2$ can you construct all solutions $(a,b)$ where $c,f$ is intenger I made an edit since the ...
0
votes
3answers
194 views

Matrix $A$ such that for all matrices $B$ the product $AB$ has a row with not a single zero

Let $A$ be a given fixed $n \times m$ matrix. We also consider matrices $B$ of dimension $m \times p$. I am interested in those matrices $A$, for which for all $B \in \mathbb R^{m \times p}$ with ...
2
votes
2answers
175 views

on the determination of a quadratic form from its isotropy group in char. 2

So this question is a continuation of the following one [1] On the determination of a quadratic form from its isotropy group For some motivations and relevant backgrounds related to this question ...
0
votes
0answers
28 views

Cholesky decomposition of a large covariance matrix

I have a tricky problem concerning a covariance matrix cholesky decomposition. What I need is to obtain the cholesky decomposition of the estimated variance matrix of the set of samples stored in a ...
4
votes
1answer
445 views

Surprising connection between linear algebra and graph theory

What is the intuition for linear algebra being such an effective tool to resolve questions regarding graphs? For example, one can determine if a given graph is connected by computing its Laplacian ...
0
votes
0answers
31 views

A Optimization problem using co-ordinates of joint numerical range.

Let $\mathbf{A}_1,\dots,\mathbf{A}_L$ be $N\times N$ hermitian matrices. Define the mapping from the $N-$dimensional unit sphere to $\mathbb{R}^L$ as \begin{align} ...
9
votes
2answers
181 views

A differentiable one-parameter family of codimension 2 subspaces of $\mathbb{C}^n$ cannot fill $\mathbb{C}^n$, right?

Suppose that $P(t)$ is a one-parameter family of rank 2 self-adjoint projections on $\mathbb{C}^n$ that vary analytically in the real parameter $t \in [0,1]$. I claim that there must exist a vector $x ...
8
votes
1answer
197 views

Trouble with Jordan form of the truncated Carleman-matrix for $\sin(x)$ as size $n$ goes to infinity

I'm currently trying to get familiar with the Jordan normal form for matrices; and after some example I ask for the possible Jordan-form for the Carleman matrix for the function $f(x) = \sin(x)$ when ...
1
vote
0answers
15 views

the 3th and 4th order statistics of Circularly Symmetric Complex Normal random vector?

Assume that ${\bf{z}} \in {\mathbb{C}}^{n \times 1}$ is a CSCG random vector denoted with $\mathcal{C} ~ (\bf{\mu} _0,\bf \Sigma _0)$ where $\mu _0$ and $\bf \Sigma _0$ are mean and contrivance ...
2
votes
1answer
137 views

Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if $$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus ...
0
votes
1answer
320 views

eigen-decomposition solution? is it unique?

Assume an N*N covariance matrix (Q) which is a positive definite matrix. The decoder X is assumed to be N*s, where s<=N. X is calculated to be s eigenvectors corresponding to s minimum eigenvalues. ...
16
votes
0answers
498 views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
0
votes
1answer
67 views

Collecting terms of a linear expression with nested sums and combinatorics in coefficients

I need to collect the $\Pr(\cdot)$ terms of the following expression: $\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left( 1-\theta \right) }\right) ^{m}}\left[ ...
0
votes
2answers
99 views

Matrix equation solving guidelines [closed]

Does anyone how to solve this matrix equation? $$ PXQ^T+P^TXQ=A$$ where all matrices are real and square. Can you provide me with some guidelines? Thanx
1
vote
0answers
68 views

What is classified by $H^1(\mathbb{R},SO(p,q))$ and by $H^1(\mathbb{R},SU(p,q))$?

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space ...
13
votes
1answer
234 views

A possible extension of a determinant inequality

It is well known that if $A, B$ are positive semidefinite matrices, then $$\det (A+B)\ge \det A+\det B.$$ I am considering a possible extension of this result. Let $\mathbb{M}_m(\mathbb{M}_n)$ ...
4
votes
1answer
743 views

Closest 3D rotation matrix in the Frobenius norm sense

Given a 3 by 3 matrix $M$ I would like to find the rotation matrix $R$ minimizing the Frobenius norm: \begin{equation} \|R-M\|_F \end{equation} Is there a closed form solution for $R$, or is it ...
6
votes
1answer
193 views

On an inequality among determinants

For Hermitian matrices $X, Y$, I write $X\ge Y\ge 0$ to mean $X-Y$ and $Y$ are positive semidefinite. In Lemma 2.5 of [Linear Algebra Appl. 452 (2014) 1-6] I proved that if $X + Y\ge W + Z$, $X\ge ...
7
votes
2answers
394 views

Elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$

For some research work, I need to know the classification of elements of finite order of $\mathrm{PGL}(n,\mathbb{Q})$, up to conjugation. Since I essentially need $n\le 4$, I think that I can show it ...
0
votes
1answer
120 views

Under what conditions there is a one-to-one mapping between a product of matrices and the sequence of matrices leading to the product?

I have a set of matrices $A_1,\ldots,A_n$. Let $\mathcal{A} = \{A_i\}$. What are some simple conditions under which for any sequence of indices between $1$ and $n$, $a_1,\ldots,a_m$, the product ...
-1
votes
0answers
15 views

2-norm of a canonical Jordan form and spectral radius [migrated]

Let J be a real square matrix which has a canonical real Jordan form. Is it true that the 2-norm of J is equal to its spectral radius? P.S.: The 2-norm of J is ...
0
votes
1answer
159 views

Find the following transformation $G$

I asked this question 6 days ago on math.stackexchange.com (http://math.stackexchange.com/questions/656585/find-the-following-transformation-g). I didn't get any answers yet, so I'm posting here. I'm ...