# Tagged Questions

Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to [tag:linear-algebra]). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur ...

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### concentration for eigenvectors

I am interested in bounding from above the ratio between the maximum and minimum entries of a Perron vector. The only results that I found in the literature are from the classic masters (Ostrowski and ...
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Does anyone have an idea how to prove the following identity? $$\mathop{\mathrm{Tr}}\left(\prod_{j=0}^{n-1}\begin{pmatrix} x^{-2j} & -x^{2j+1} \\ 1 & 0 \end{pmatrix}\right)= \begin{cases} ... 2answers 3k views ### Determinants in Graph Theory In graph theory, we work with adjacency matrices which define the connections between the vertices. These matrices have various properties in themselves. For example, their trace can be calculated (it ... 0answers 754 views ### eigenvalues of a symmetric tridiagonal matrix with zero diagonals I was investigating a problem and came up with the following symmetric tridiagonal matrix (with zero diagonal elements):$$ \left(\begin{array}{cccccc} 0 & a & 0 & \ldots & 0 \\ a ...
Suppose: $B_i \in \mathbb{C}^{n \times n}$, $0<w_i\in \mathbb{R}$ $(i = 0,1,2,\ldots,m)$ ${\rm P}(x) ={\rm{B}_m} x ^m + \cdots + B_1 x + B_0$ is a matrix polynomial, and $x$ is a complex ...