Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials) \begin{equati …

Let $\mathcal{R}$ be a ring and let $v^0,\ldots,v^{k-1}\in\mathcal{R}^m$ with $m \geq k$. Suppose we wish to find $w\in Span(v^0,\ldots,v^{k-1})$ such that $k-1$ specified coordina …

I am trying to prove that a mapping has a unique fixed-point by showing that its Jacobian is a P-matrix. In this particular case the Jacobian can be decomposed as the sum of two ma …

Hi all I have a matrix $\Sigma$ with element $(i,j)$ $\Sigma_{i,j}= exp(-h_{i,j}\rho)$. The matrix is positive definite and symmetric (it is a covariance matrix). Now i need to …

For any $n$ distinct points $x_1,x_2 , \ldots , x_n$ on the real line show that the matrix $M$ where $M(i,j) = e^{\lambda_j x_i} $ has non-zero determinant where $\lambda_1 \lt \ …

Introduction: As a quick reminder, the Gel'fand Yaglom theorem uses the generalized zeta-function approach to compute functional determinants of differential operators. Given a di …

Let ${p_n}(x) = x{p_{n - 1}}(x) - {a_{n - 2}}{p_{n - 2}}(x)$ for some numbers ${a_n}$ with initial values ${p_{ - 1}}(x) = 0$ and ${p_0}(x) = 1.$ By Favard’s theorem about ortho …

The two most elementary ways to prove an N x N matrix's determinant = 0 are: A) Find a row or column that equals the 0 vector. B) Find a linear combination of rows or columns tha …

This might be ridiculously obvious, but... For each $n \in \mathbb{N}$, let $M_n$ denote the manifold of $n \times n$ matrices with real entries. It is well known that the $n$-dim …

Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$. And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and …

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to comput …

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences. As …

In June 2012, Bill Press and Freeman Dyson published a remarkable paper on the iterated prisoner's dilemma. A key step in their derivation is a simple fact from linear algebra that …

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity $$ \det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \de …

Context: some probably know that there are Capelli identities which state det(A)det(B) = det(AB+correction) for some matrices with non-commuting elements, they go back to 19-th cen …