# Tagged Questions

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### Determinantal formula for the nullspace of a singular matrix

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 I feel I should ...
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### a determinantal identity

Dusan Pokorny and Jan Rataj have just posted a paper (http://arxiv.org/abs/1209.2305) in which they prove the identity $$\det (A-B) = \frac 1{d!} \sum_{k=0}^d (-1)^k \binom dk \det((d-k)A + kB)$$ ...
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### 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?
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### Generalizations of Oppenheim's inequality

The well-known Oppenheim inequality says that for two positive definite matrices $A,B$ it holds that $\det(A \circ B) \geq (\prod{a_{ii}})\det(B)$. There has been a lot of beautiful work done ...
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### Is there a simple relation between the entropy of a matrix and its characteristic polynomial?

Assume $M$ is an invertible positive matrix of rank $N$. The Von Neumann entropy $H$ of a matrix $M$ with eigenvalues $\{ \lambda_n \}$ is $H[M] = -\sum_{n=1}^N \lambda_n \ln \lambda_n$. In ...
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### determinantal identity sought

Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I_{n}+AB)=det(I_{m}+BA)$. Is there an analogous identity of the form $det(P_{1}+AB)=det(P_{2}+BA)$, ...
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### Extremal properties of the determinant for matrices with entries in a fixed subset of $[-1,1]^{n^2}$?

Given a multiset $S\subset [-1,1]^{n^2}$, we set $$m(S)=\min\vert \det(M)\vert$$ where the minimum is over all matrices with entries forming the multiset $S$ and $$a(n)=\max m(S)$$ where the maximum ...
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### Counting matrices with different determinants

Let $A$ and $B$ be two matrices of order $n$ over a finite subset of integers $S$ such that $A$ and $B$ are positive-definite, nonsingular and symmetric. I am interested in proprieties about $A$, $B$ ...
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### Encoding information about submatrix determinants

$M$ is an $n\times n$ matrix. Consider the submatrices $M(P;Q)$ formed from $P$ rows and $Q$ columns of $M$ where $P$ and $Q$ are disjoint indices. Is there some way to encode the various ...
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### Question on a relation between minors of a particular kind of matrix

Hi! Perhaps it is an easy question but i don't figure out how to prove it. Let $a_1,...,a_{2m+2}\in\mathbb{C}$ and for $1\leq i\leq 2m+2$ and $j\leq [\frac{2m+2-i}{2}]$ (with $[a]$ i mean the integer ...
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### Determinant and symmetric power

Let $V$ be a vector space over some field $k$ and $T \in \mathrm{GL}(V)$. Then, we can view $T\in \mathrm{GL}(\mathrm{Sym}^k(V))$ where $\mathrm{Sym}^k(V)$ denotes the $k^\mathrm{th}$ symmetric power ...
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### What's the maximum determinant of the (0, 1) matrix from M(n, R)?

If there's no exact formula what's the nearest upper and lower bounds do you know?
The determinant behaves multiplicatively with respect to the usual matrix product $$\det(AB) = \det(A)\det(B),$$ and also with respect to the Kronecker (or tensor) product of square matrices $$... 0answers 764 views ### Determinant of a sum of a diagonal matrix, a dyadic product matrix, and a Hermitian Toeplitz matrix Hi From a physics problem, I am trying to evaluate exactly the following kind of determinant: G = A + M + N. A is diagonal M is a product of a column (of 1s) and a row matrix N is a Hermitian ... 19answers 16k views ### Why were matrix determinants once such a big deal? I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ... 3answers 2k views ### Splitting the determinant polynomial into linear factors - a Dedekind problem Here's the question in a nutshell. For some n\in\mathbb N, we consider the polynomial \det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq ... 3answers 1k views ### Sarrus determinant rule: references, extensions SEEKING REFERENCES FOR SARRUS' RULE AND EXTENSIONS An undergraduate came to me with an identity for 4x4 determinants that is actually correct: \det(A)=h(A)+h(RA)+h(R^{2}A) where R cyclically ... 2answers 1k views ### Statement of Lagrange's theorem on determinants(elementary question). Apologies for this elementary question; but I was unable to find a reference otherwise. Let A, B, C be square matrices of the same dimension. Then,$$\begin{vmatrix} A & C \\\ 0 & B ...
By a pairing on a vector space $V$, I mean a linear map $A : V \otimes V \to R$. If $V$ is $n$-dimensional ($n < \infty$), then I can define the determinant of $A$ by considering the canonical ...