Search Results

This is (quite obviously) inspired by this question. Let $C_i$ be symmetric positive definite matrices. Then is it true that there is exactly one symmetric positive definite $X$ such that $F(X) = X^n …

If you have two linear subspaces $V_1$ and $V_2$ of a vector space $V,$ both given by their bases, there is fairly heavy handed way of computing their intersection: write down the projection matrices …

This may be a really dumb question, but here goes: is there any algorithm to compute the signature of a quadratic form (or a symmetric matrix, if you prefer) more efficient (asymptotically or otherwi …

Suppose I have a finite subset $\mathcal{M}$ of a Banach space $B$ $\mathcal{M}=p_1, \dots, p_n,$ and I create the following ``Gram'' matrix $G_{\mathcal{M}}:$ $$g_{ij} = \frac{\|p_i\|^2 + \|p_j\|^2 …

Suppose I have a Smith normal form $S,$ and I want to have an $M \in SL(n, \mathbb{Z}),$ such that $M - I$ has SNF $S.$ Is this always possible? For a (potentially) somewhat harder question, what if I …

Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does …

I have a feeling that this might have already been asked, but can't find the question. Anyway, the question is: given a symmetric $n\times n$ matrix, is there a faster way to compute its signature tha …

Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of …

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) …

Consider the Haar measure $\mu$ on the Grassmannian $G(n, k)$ of $k$-dimensional subspaces in $\mathbb{R}^n.$ Now, pick pairs of subspaces uniformly at random with respect to $\mu,$ and compute their …

This is related to this question. Suppose I have an $n$-dimensional representation of a finitely generated group, and I want to know whether it is absolutely irreducible. This can, of course, be done …

Take your favorite matrix norm $\|\bullet\|$ (my favorite is the Frobenius norm $\|A\| = \sqrt{\operatorname{tr} A A^t}$). Now consider the set $S_x$ of matrices $A,$ such that $\|A\| < x$ and $\|A^{ …

To be concrete, let $G=SL(n, \mathbb{C}),$ $\phi$ an automorphism of $G.$ Is there a characterization of those $x$ for which there exists a $y$ such that $x = y \phi(y)?$ In the special case, the auto …

Suppose you have a matrix $M$ in $SL(n, \mathbb{Z}).$ Question: is there a necessary and sufficient condition for $M$ to be conjugate to $N \in Sp(n, \mathbb{Z}).$ It is clearly necessary that the cha …

In this thesis, Pedro Freitas discusses the properties of distance functions on matrices defined by $d_p(A, B) = (\sum (\log (\sigma_i(A^{-1} B)))^p)^{1/p}.$ Here $\sigma_i$ are the singular values of …