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7
votes
1answer
457 views

Does this Linear Algebra Construction have a Name?

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 coordinates of $w$ vanish ...
9
votes
0answers
478 views

slick-proof-of-trick-for-counting-domino-tilings

The trick for rewriting the number of domino tilings of a simply-connected finite lattice region as the absolute value of the determinant of a matrix (due I believe to Kasteleyn and Percus, but if ...
6
votes
0answers
454 views

Can one give a “nice” expression for this determinant?

I am asking this question on behalf of a senior faculty member who is sometimes intimidated by computers. It is motivated by a problem in invariant theory. Unfortunately the question is a bit vague. ...
5
votes
0answers
321 views

Determine if a matrix is unimodular

Is deciding if an integer square matrix has determinant $\pm 1$ faster that calculating the determinant of the matrix?
5
votes
0answers
369 views

Characteristic polynomials of certain random symmetric matrices and the complexity of random Morse functions

Investigations concerning random Morse functions led me to the following problem. Consider the classical GOE of $m\times m$ real symmetric matrices $A$ with independent Gaussian entries with ...
3
votes
0answers
78 views

Infinite series of determinants

I am interested in what is known about the following class of sums. For a sequence of matrices $A_i$ (which possibly have different size), I am wondering about examples and methods for evaluating sums ...
3
votes
0answers
415 views

determinant of fibonacci-sum graphs

We have a simple graph with vertices $\{v_1, v_2, ... v_n\}$. The adjacency matrix of this graph is $A= (a_{ij})$ so that $a_{ij}=1$ if $i+j$ belongs to the Fibonacci sequence; $a_{ij}=0$ ...
3
votes
0answers
236 views

det(A)det(B) = det(AB+correction), Capelli identities, “factorzied” representation of gl_n

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 century, but also ...
3
votes
0answers
706 views

Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix

Hi From a physics problem, I am trying to evaluate exactly the following kind of determinant: G = A + M + N. A is diagonal M is a product of a column (of 1s) and a row matrix N is a Hermitian ...
2
votes
0answers
267 views

Morphisms of Spectral Sequences and alternating products

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. Assume that morphisms ...
2
votes
0answers
121 views

which deformation of a matrix lead to flat deformations of determinantal varieties (fitting ideals)?

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, ...
2
votes
0answers
246 views

Function recursion relation over symmetric group

Hi! Let P be a permutation in the symmetric group SN and let π=πj, j+1 be a transposition of elements j and j+1 of the permutation. Let A(P) be a function in dependence of the permutation P. ...
1
vote
0answers
58 views

range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants

The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
1
vote
0answers
124 views

Polynomials satisfying a three-term recurrence

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 orthogonal polynomials ...
1
vote
0answers
150 views

Generalized Schur polynomial from block Toeplitz matrices

By using the Jacobi-Trudi identity, one may interpret banded Toeplitz matrices, and minors of such matrices in terms of Schur polynomials, see for example ...
0
votes
0answers
154 views

Inertia/Gravity in Distance Geometry

The Cayley-Menger Determinant, D(N), slickly calculates the N-dimensional simplex volume of any N+1 points. One constraint in our 3D world is that D(4)=0. Give each point a mass (Mi) and dynamic ...