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Results tagged with matrix-theory
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user 91764
Matrix theory is the study of matrices as concrete objects, rather than as abstract linear operators between vector spaces (whose study belongs to linear algebra). For instance, this involves matrix factorizations and decompositions, nonnegative matrices and Perron-Frobenius theory, Schur complements, structured and special matrices, matrix functions and equations.
4
votes
Solving $\text{trace}\left[\left(I + pY\right)^{-1} \left(I - p^{2}Y\right)\right] = 0$ for ...
Given $\mathrm A \in \mathbb R^{n \times n}$, we define $f : \mathbb R \to \mathbb R$ as follows
$$f (x) := \mbox{tr} \left( (\mathrm I_n + x \mathrm A)^{-1} (\mathrm I_n - x^2 \mathrm A) \right)$$
…
10
votes
Accepted
Decomposing a matrix into a product of sparse matrices
Given an invertible $n \times n$ matrix $\mathrm A$, we perform Gaussian elimination until we obtain a (nonsingular) diagonal matrix. In other words, we left-multiply $\mathrm A$ by permutation matric …
1
vote
Behavior of matrix rank under thresholding of its elements
In some cases, the rank is preserved under thresholding. For example, let
$$\rm A := \begin{bmatrix} 1\\ 0\\-1\end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix}^\top = \begin{bmatrix} 1 & 1 & 1\\ 0 …
1
vote
Submatrix with small sum of elements
Given
a positive $n \times n$ real matrix $\mathrm A$
a positive integer $m < n$
we would like to select $m$ rows and $m$ columns of $\mathrm A$ such that the sum of the (positive) entries of the se …
4
votes
Nontrivial lower bound on the sum of matrix norms
Let function $f : \mathbb R^{m \times n} \to \mathbb R_0^+$ be defined as follows
$$f (\mathrm X) := \| \,\mathrm X \mathrm A^\top \|_\text{F}^2 + \| \,\mathrm X^\top \mathrm A \,\|_\text{F}^2 + \lef …