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Results tagged with matrix-analysis
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user 91764
The study of the properties of real and complex matrices that are more close to analysis and operator theory. For instance: the properties of positive definite matrices, matrix inequalities, perturbation analysis, matrix functions, inequalities between eigenvectors and singular values, majorization.
1
vote
Minimizing $\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$ with respect to matrix $X$
Given $\mathrm X_0, \mathrm Y_0 \in \mathbb R^{n \times n}$,
$$\begin{array}{ll} \text{minimize} & \|\mathrm X - \mathrm X_0\|_F^2 + \|\mathrm Y - \mathrm Y_0\|_F^2\\ \text{subject to} & \mathrm Y = …
1
vote
Minimizing $\|X-X_0\|_F^2+\|X^{-1}-Y_0\|^2_F$ with respect to matrix $X$
Given $\mathrm X_0, \mathrm Y_0 \in \mathbb R^{n \times n}$,
$$\begin{array}{ll} \text{minimize} & \|\mathrm X - \mathrm X_0\|_F^2 + \|\mathrm Y - \mathrm Y_0\|_F^2\\ \text{subject to} & \mathrm Y \m …
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
$$f (x) := \mbox{tr} \left( (\mathrm I_n + x \mathrm A)^{-1} (\mathrm I_n - x^2 \mathrm A) \right)$$
…
9
votes
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
Some call them currency exchange matrices. From Boyd & Vandenberghe's Introduction to Applied Linear Algebra:
6.7 Currency exchange matrix. We consider a set of $n$ currencies, labeled $1,\dots,n$. ( …
1
vote
Behavior of matrix rank under thresholding of its elements
In some cases, the rank is preserved under thresholding. For example, let
$$\rm A := \begin{bmatrix} 1\\ 0\\-1\end{bmatrix} \begin{bmatrix} 1\\ 1\\ 1\end{bmatrix}^\top = \begin{bmatrix} 1 & 1 & 1\\ 0 …
5
votes
Is this inequality involving the Frobenius norm right?
Given $\mathrm A \in \mathbb R^{m \times n}$ and $\mathrm B \in \mathbb R^{n \times p}$, let $\mathrm B \mathrm B^{\top} = \mathrm Q \Lambda \mathrm Q^{\top}$ be an eigendecomposition of $\mathrm B \m …
3
votes
Determinant of a $k \times k$ block matrix
Let us assume that $A-B$ is invertible. Write
$$\begin{array}{rl} C &= \begin{bmatrix} A & B & \ldots & B\\ B & A & \ldots & B\\ \vdots & \vdots & \ddots & \vdots\\B & B & \ldots & A\end{bmatrix}\\\\ …
14
votes
Determinant of a $k \times k$ block matrix
Subtracting the last row of blocks from the first $k-1$ rows of blocks, we obtain
$$\begin{bmatrix}A-B & O & O & \dots & O & B-A\\ O & A-B & O & \dots & O & B-A\\ O & O & A-B & \dots & O & B-A\\ \vdo …
4
votes
Nontrivial lower bound on the sum of matrix norms
Let function $f : \mathbb R^{m \times n} \to \mathbb R_0^+$ be defined as follows
$$f (\mathrm X) := \| \,\mathrm X \mathrm A^\top \|_\text{F}^2 + \| \,\mathrm X^\top \mathrm A \,\|_\text{F}^2 + \lef …