# Tagged Questions

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### Dimension of a similarity class

Let $K$ be an algebraically closed field with characteristic $0$. I consider the Jordan decomposition of a NILPOTENT matrix: $A=diag(J_{r_1},\cdots,J_{r_s})\in M_n(K)$ where $J_k$ is the nilpotent ...
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### dimensions of strata of Pfaffian varieties

Let $V$ a complex vector space of dimension $2n$. Let us consider $W=\wedge^2V$ and the Pfaffian variety $Pf\subset \mathbb{P}W$ that parametrize degenerate skew-symmetric matrices. $Pf$ is naturally ...
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### Simultaneous triangularizability over a commutative ring

Let $R$ be a commutative ring with unity and $A,B\in M_n(R)$ satisfying the property (*) All elements of the two-side ideal, in $M_n(R)$, generated by $AB-BA$, are nilpotent. McCoy showed that, if ...
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### Vector Spaces of Symmetric Matrices of Low Rank

Let $K$ be a field, if necessary algebraic closed or of characteristic zero. Let $k$ be a positive integer. I am interested in linear subspaces $M \subseteq \textrm{Sym}_n(K)$, where ...
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### Grassmann-Plücker relations for permanents

Let $K$ be a field, $1 \leq d \leq n$ integers and $V$ an $n$-dimensional vector space. The Grassmann-Plücker relations are quadratic forms on $\wedge^d V$ whose zero set is exactly the set of ...
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### Lie's Theorem in characteristic $p$

Let $K$ be an algebraically closed field with characteristic $0$ and $V$ be a Lie sub-algebra of $M_n(K)$, the $n\times n$ matrices over $K$. If $V$ is solvable, then, according to Lie's theorem, $V$ ...
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### Solving $P=AB,Q=BA$, in the unknowns $A,B$

Let $p\geq q$ $P\in M_p(\mathbb{C}),Q\in M_q(\mathbb{C})$. We seek $A\in M_{p,q},B\in M_{q,p}$ s.t. $P=AB,Q=BA$. The NS conditions for the existence of $(A,B)$ are given in On the matrices AB and BA. ...
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### Dimension of the nilpotent centralizer of a nilpotent matrix

Fix a natural number $n$ and an algebraically closed field $k$. Let $\mathfrak{g}=\mathfrak{gl}_n(k)$. For any partition of $n$, $\lambda=(\lambda_1,\ldots,\lambda_r)$, let $A_{\lambda}$ be the ...
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Consider $S$, the set of all $n\times m$ real matrices with specified row sums $(r_1,...,r_n)$, column sums $(c_1,...,c_m)$, and strictly positive entries. For any matrix $A$, define $$... 1answer 151 views ### A natural bijection between the orbit spaces of p-nilpotent matrices for varying p Let k be an algebraically closed field of characteristic p, call a matrix X\in\mathfrak{gl}_n(k) p-nilpotent if X^p=0, and let \mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k)) be the set of ... 1answer 201 views ### differential of the characteristic polynomial Let \chi:GL_{n}(\mathbb{C})\rightarrow \mathbb{C}^{n} the map given by the coefficients of the characteristic polynomial. Let A a regular semisimple matrix, do we have a formula for the ... 5answers 663 views ### Does this matrix shape have a name? I'm using a lot of matrices that look like this:$$A_3 = \begin{bmatrix} a & b & b\\ b & a & b\\ b & b & a \end{bmatrix} $$i.e. the diagonal entries are all the same, and all ... 2answers 842 views ### On the Positive Definiteness of a Linear Combination of Matrices In my work in PDE, the following problem in linear algebra came up. Any help in this direction is appreciated. QUESTION: Let m,n\in\mathbb{N} and let A_1,\ldots, A_m\in M_n(\mathbb{R}) be real, ... 3answers 797 views ### Approximating commuting matrices by commuting diagonalizable matrices Suppose the matrices A and B commute. Do there exists sequences A_n and B_n of matrices such that A_n \rightarrow A, B_n \rightarrow B. Each A_n is diagonalizable and the same for ... 2answers 361 views ### Determinantal rings are Cohen-Macaulay Consider a n\times n matrix M with entries in R=\mathbb{C}[x_1,\dots,x_n]. Let I be the ideal of (n-1)\times(n-1) minors of M. Is \mathcal{O}_{\mathbb{C}^n}/I Cohen-Macaulay?If not, ... 3answers 321 views ### Linear subspaces in cones over orthogonal groups Consider the orthogonal group G=O(n) as a subset of the vector space of n\times n real matrices. Let C=C(G) denote the Euclidean cone over G, i.e., the space of matrices of the form tA, A\in ... 2answers 255 views ### Smooth a matrix I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you ... 1answer 442 views ### existence of polynomial equation system solution For 1 \leq i \leq n, let A=\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & \ddots & \vdots \\ a_{n1} & \cdots & a_{nn} \\ \end{bmatrix} B_i=\begin{bmatrix} b_{i1} ... 0answers 819 views ### Cartan decomposition for upper triangular matrices Due to the comments, I have the impression that I have to be more precise. Consider G= GL_n(F) for a non-Archimedean field F with ring of integers o. Let K= GL_n(o) and let I the Iwahori ... 1answer 1k views ### Constructing a unitary matrix Setting: Given a set of n\times n matrices A_i, I would like to find a linear combination of these matrices Q = \sum_i A_i x_i with x_i a set of complex numbers, such that Q is unitary: ... 2answers 408 views ### The topology of open semi-algebraic sets (appl.: totally positive matrices) Let P_1,\ldots,P_r be polynomials over {\mathbb R}^N. I am interested in the homotopy type of the semi-algebraic set defined by$$ P_j(x_1,\ldots,x_N)>0,\qquad j=1,\ldots,r . $$Is there a ... 3answers 646 views ### Detecting if a polynomial is a Pfaffian Given an explicit polynomial, is there any kind of trick/algorithm to check whether it is a pfaffian of a matrix with linear entries? 4answers 2k views ### Irreducibility of determinant of symmetric matrix It is quite known fact that the determinant of arbitrary symmetric matrix is an irreducible polynomial in algebra \mathbb C [x_{ij}, 1\leq i,j\leq n] (x_{ij}=x_{ji}) (see this: ... 0answers 130 views ### Standard polynomials applied to matrices (bis) The standard polynomial in r non-commuting indeterminates x_1,\ldots,x_r is defined by$${\mathcal S}_r(x_1,\ldots,x_r):=\sum_{\sigma\in S_r}\epsilon(\sigma)x_{\sigma(1)}x_{\sigma(2)}\cdots ...
Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish. I remember hearing that one really does not need to check all possible minors in order to conclude ...
A quaternionic matrix $A$ gives rise to a function $\mathbb{H}^n \to \mathbb{H}^n$ given by $x \mapsto A \cdot x$. This is real linear, but not complex- or quaternionic-linear (in general) if we ...