# Tagged Questions

For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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### How do I find the common invariant subspaces of a span of matrices?

Let $G_1, \ldots, G_n$ be a set of $m\times m$ linearly-independent complex matrices. Let $\mathcal{G} = \operatorname{span}\left\{ G_1, \ldots , G_n\right\}$ be the vector space that spans the set ...
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### Representation for Vandermonde's permanent

Permanent of a matrix A = $\|a_{i,j}\|_{i,j=1}^{n}$ is defined as $$\mathrm{Perm}(A) = \sum\limits_{\sigma \in S_{n}} a_{1,\sigma_{1}},\ldots,a_{n,\sigma_{n}}$$ Is there some representation for ...
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### Hadamard matrices and sub-matrices (Converse of Sylvester Construction)

Let $H$ be a $d$ by $d$ real Hadamard matrix, namely: $$HH^{T}=d I$$ where $I$ is the identity matrix and $d=2^{k}$ for some natural number $k\geq 2$. The entries of $H$ are either $1$ or $-1$ and it ...
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### Exactly $n-1$ nonzero elements if $\det(A)=0$ for every arrangement

Let $x_1,x_2,\dots,x_{n^2}\in\mathbb{R}$ with the property that any $n\times n$ matrix with exactly these elements has determinant $0$. Suppose also that there are at least $n$ distinct elements. ...
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### Proof of the conjecture that the kernel is of dimension 2, extended

Pursuing my research, I am now looking for a proof of an extension of the problem proposed here and answered. It's an extension in the sense that I'm now considering two different $t_1$ and $t_2$. The ...
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### Is there any specific criteria for a matrix $A(t)$ to have either time-dependent or independent eigenvectors?

I am investigating properties of a matrix $$A(t_1,t_2) \equiv U_1(t_1) \otimes U_2(t_2) - U_2(t_2) \otimes U_1(t_1)$$ where $U_1$ and $U_2$ are time-dependent unitary matrices. I'm finding that for ...
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### Hyperdeterminant of 4x4x4 hypermatrix

If given the hypermatrix (which I've written here in bracket notation since I'm not all too sure how to display this) { {{1,1,1,1},{1,1,-1,-1},{1,-1,-1,1},{1,-1,1,-1}}, {{1,1,1,1},{1,1,-1,-1},{1,-1,1,-...
Letting $n \in \mathbb{N}$, let $M_{n}$ denote the $n \times n$ binary matrix with ones along the main antidiagonal and everywhere below the main antidiagonal and ones along the antidiagonal two ...