**17**

votes

**3**answers

885 views

### Example of a space for which $V \cong Hom(V,V)$

Let $V$ be a topological linear space, and let $\operatorname{Hom}(V,V)$ be the space of continuous linear maps from $V$ back to $V$, equipped with a suitable topology.
Is there a non-trivial ...

**3**

votes

**0**answers

14 views

### Factor a sum of products of cofactors

Let $M$ be an $n\times n$ matrix whose first column consists entirely of 1s.
We define the usual cofactors: $C_{i,j}$ is $(-1)^{i+j}$ times the determinant of the submatrix obtained by deleting row ...

**6**

votes

**3**answers

219 views

### Diagonalization via the Toda flow

According to some almost indecipherable notes of a graduate Linear Algebra class, a symmetric matrix $A\in\mathbb R^{n\times n}$ can be diagonalized via the Toda flow. More specifically, if ...

**8**

votes

**2**answers

467 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

**1**answer

160 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
...

**0**

votes

**1**answer

172 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 ...

**0**

votes

**0**answers

64 views

### Isomorphic Dual and Conjugate Representations of a Lie Algebra

Let $\frak{g}$ be a complex Lie algebra $\frak{g}$, and $R:\frak{g} \to $End$(V)$, a representation for some finite dimensional complex vector space $V$. As is well-known, we can construct from $R$ ...

**10**

votes

**2**answers

303 views

### 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 ...

**-2**

votes

**0**answers

32 views

### multiplication of a projection matrix and PSD matrix is a PSD? [on hold]

I have a projection matrix P and X^TAX where A is a diagonal matrix with all strictly positive entries can I tell that PX^TAX is PSD?

**1**

vote

**0**answers

87 views

### Default Orientation of Vectors [on hold]

When I started studying math in 1982 in Germany, there seemed to have been a change in the choice of the default orientation of vectors; while it was row-vectors till then, it changed to ...

**6**

votes

**1**answer

160 views

### Horn's inequalities for n matrices

Where I can find necessary and sufficient conditions on eigenvalues of Hermitian matrices with the relation $$A_1 + A_2 + ... + A_n = A_0 ,$$
i.e. Horn's inequalities for n matrices?
Can such ...

**8**

votes

**1**answer

240 views

### Why does this antisymmetric product factor out a determinant?

Consider a generic $n \times n$ matrix $M$.
Define the $(n-1) \times n$ matrix $M_q$ to be $M$ with the $q$th row omitted, and assume that $M_q$ possesses a right inverse, $R_q$:
$$R_q = M_q^T (M_q ...

**2**

votes

**2**answers

89 views

### Boundedness of ratio of linear functions

Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...

**3**

votes

**2**answers

60 views

### Norm of triangular truncation operator on rank deficient matrices

Let $T_{n\times n}$ be a triangular truncation matrix, i.e.
$$T_{i,j}=\begin{cases}1 & i\ge j\\ 0 & i<j \end{cases}$$
It is known that for arbitrary $A_{n\times n}$
$$\|T\circ ...

**2**

votes

**0**answers

40 views

### Reducing $\ell_1$ norm of non-full-rank matrices

I have two matrices ${\bf{X}}_{p\times r}$ and ${\bf{Y}}_{r\times q}$ with $r<\min(p,q)$. Matrix ${\bf Y}$ does not have full row rank (i.e., rank$({\bf Y})<r$). Can I build two new matrices ...

**-1**

votes

**0**answers

124 views

### Why do I study a lot but still don't understand the material too well? [on hold]

I had a linear algebra test yesterday and I studied a week in advance for it. I did all the assigned homework questions, past exam questions, problem set questions, but I still did poorly on the test. ...

**-4**

votes

**0**answers

24 views

### Help with simple rotation on an x,y plane [migrated]

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane.
Something that is hopefully quite easy for you guys. ...

**3**

votes

**3**answers

2k views

### Derivative of a determinant of a matrix field

Let $A(x_1,...,x_n)$ be an $n\times n$ matrix field over $R^n$.
I am interested in the partial derivative determinant of $A$ in respect to $x_i$. In can be shown that:
...

**3**

votes

**1**answer

102 views

### What is a degenerate Legendre Transformation?

I am studying the Lagrangian and Hamiltonian description of some dynamical systems. The problem with this description of the particular kind of systems I am studying, is that the Legendre ...

**1**

vote

**1**answer

130 views

### Linear map with two “incompatible” representations

Let $K$ be a field and let $V$ be the set of sequences $\{v_1,v_2,\dots\}$ of elements of $K$. If $A=\{a_1,a_2,\dots\}$ is also a sequence of elements of $K$, then it defines an endomorphism of $V$ ...

**2**

votes

**1**answer

189 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 ...

**0**

votes

**0**answers

40 views

### Uniqueness of a quadratic time-dependent matrix equation

Let $v: [0,1] \to \mathbb R^n, t \mapsto v(t)$ continuously differentiable with the property that for any constant vector $h \in \mathbb R^n$ the fact that $v(t)^{\top} h = 0$ for all $t \in [0,1]$ ...

**-6**

votes

**0**answers

48 views

### solve for three unknowns. [closed]

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

**1**answer

107 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

**0**answers

224 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

**1**answer

41 views

### NonLinear Maps and homogeneity [closed]

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

**2**answers

160 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

**3**answers

202 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

**1**answer

121 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

**7**

votes

**1**answer

140 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

**2**answers

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

**2**answers

699 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 ...

**2**

votes

**0**answers

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

**1**answer

307 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

**10**answers

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

**1**answer

84 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

**1**answer

95 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

**4**answers

186 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

**1**answer

65 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

**3**answers

531 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

**16**answers

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

**1**answer

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

**2**answers

3k 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

**1**answer

121 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

**1**answer

100 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

**0**answers

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

**3**answers

197 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

**2**answers

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

**0**answers

32 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

**1**answer

453 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 ...