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Results tagged with linear-algebra
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user 91764
Questions about the properties of vector spaces and linear transformations, including linear systems in general.
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
$$\begin{bmatrix}A-B & O & O & \dots & O & B-A\\ O & A-B & O & \dots & O & B-A\\ O & O & A-B & \dots & O & B-A\\ \vdo …
10
votes
Accepted
Decomposing a matrix into a product of sparse matrices
Given an invertible $n \times n$ matrix $\mathrm A$, we perform Gaussian elimination until we obtain a (nonsingular) diagonal matrix. In other words, we left-multiply $\mathrm A$ by permutation matric …
9
votes
Is there a name for matrices of the form $a_{ij}=\frac{1}{a_{ji}}$?
Some call them currency exchange matrices. From Boyd & Vandenberghe's Introduction to Applied Linear Algebra:
6.7 Currency exchange matrix. We consider a set of $n$ currencies, labeled $1,\dots,n$. ( …
7
votes
Accepted
Finding Toeplitz matrix nearest to a given matrix
The set of $n \times n$ symmetric Toeplitz matrices is
$$\left\{ x_1 \mathrm M_1 + x_2 \mathrm M_2 + \cdots + x_n \mathrm M_n \mid x_1, x_2, \dots, x_n \in \mathbb R \right\}$$
where $\mathrm M_1, \m …
6
votes
Accepted
Minimization problem involving the inverse of an affine matrix function
Rephrasing slightly, given (symmetric) matrix $\mathrm A \succeq \mathrm O_n$, we have the following minimization problem in (symmetric) matrix $\mathrm X \succeq \mathrm O_n$
$$\begin{array}{ll} \te …
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 …
5
votes
Accepted
Positive definite matrices diagonalised by orthogonal matrices that are also involutions
Let $n \times n$ matrix $\rm A$ be symmetric and positive definite. Since $\rm A$ is symmetric, it is diagonalizable. Hence, there exists a (non-singular) matrix $\rm P$ such that $\mathrm A = \mathrm …
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
$$f (x) := \mbox{tr} \left( (\mathrm I_n + x \mathrm A)^{-1} (\mathrm I_n - x^2 \mathrm A) \right)$$
…
4
votes
Determining if some permutation of a vector satisfies a system of linear equations
Let $\mathbb P_n$ be the set of $n \times n$ permutation matrices. Given matrix $\mathrm A \in \mathbb R^{m \times n}$ and vector $\mathrm v \in \mathbb R^n$, we would like to find a permutation matri …
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
$$f (\mathrm X) := \| \,\mathrm X \mathrm A^\top \|_\text{F}^2 + \| \,\mathrm X^\top \mathrm A \,\|_\text{F}^2 + \lef …
3
votes
Maximize the Euclidean norm of a matrix times a vector on unit sub-spheres
$$\mathrm A \mathrm x = \begin{bmatrix} \mathrm A_1 & \mathrm A_2 & \cdots & \mathrm A_n\end{bmatrix} \begin{bmatrix} \mathrm x_1\\ \mathrm x_2\\ \vdots \\ \mathrm x_n\end{bmatrix}$$
where $\mathrm x …
3
votes
Determinant of a $k \times k$ block matrix
Let us assume that $A-B$ is invertible. Write
$$\begin{array}{rl} C &= \begin{bmatrix} A & B & \ldots & B\\ B & A & \ldots & B\\ \vdots & \vdots & \ddots & \vdots\\B & B & \ldots & A\end{bmatrix}\\\\ …
2
votes
Integral of the entrywise square of the exponential of a matrix
Given a Hurwitz matrix $\mathrm A \in \mathbb R^{n \times n}$, let
$$\Phi (t) := \exp(\mathrm A t)$$
be the state transition matrix, and let its $(i,j)$-th entry be denoted by
$$\varphi_{ij} (t) := …
2
votes
Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Minimizing $f$ and $h$ subject to the given constraints are quadratic programs of the form
$$\begin{array}{ll} \text{minimize} & \frac 12 \mathrm x^{\top} \mathrm A \,\mathrm x\\ \text{subject to} & …
2
votes
Solving diagonal simultaneous quadratic equations
We have the following system of quadratic equations in $\mathrm x \in \mathbb R^n$
$$\mathrm A (\mathrm x \circ \mathrm x) + \mathrm B \mathrm x + \mathrm c = 0_m$$
where $\mathrm A \in \mathbb R^{m …