Tagged Questions

Questions about the properties of vector spaces and linear transformations, including linear systems in general.

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The effect of linear transformation on generic vectors [closed]

I have a question about the effect of applying a linear transformation $M$ in $\mathbb{R}^{n \times n}$ to a vector $v \in \mathbb{R^n}$. I know that if $M$ has p-norm $\|M\|_p = \lambda$, then by ...
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Two minimization problems using singular value decomposition

Posted here too: http://math.stackexchange.com/questions/1711026/two-minimization-problems-using-singular-value-decomposition Let $q_0, q_1:[0,1]\to \mathbb{R}^n$ be two maps whose components are ...
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$P(Z)$ is matrix polynomials. Why is $s_n$ smooth in a neighbourhood of $Z$?

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(Z) = A_m Z^m + \cdots + A_1 Z + A_0$ is a matrix polynomial, and $Z$ is a complex variable. $Z$ is eigenvalue of $P(Z )$ if ...
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On optimizing a function whose projection and projected vector go through a linear transformation

Assume the two sets of vectors $\{\mathbf{a}_1,\ldots,\mathbf{a}_N\}$ and $\{\mathbf{b}_1,\ldots,\mathbf{b}_N\}$ of equal length. My goal is to find the optimum matrix $\mathbf{C}$ to the following ...
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Can every trace preserving isomorphism of unital self-adjoint matrix algebras be realized as conjugation by a unitary?

In this paper, Friedland shows (in Lemma 3.4) that if $\phi$ is an isomorphism of coherent algebras, then there exists a unitary $U$ such that $$\phi(M) = UMU^\dagger$$ for all $M$. I am wondering if ...
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Unusual inequality concerning elementary symmetric functions

To state the question and fix conventions I will introduce some notation from e.g. (Lin-Trudinger, Bull. Aust. Math. Soc. 1994, On some inequalities for elementary symmetric functions") Given ...
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Eigenvalue perturbation of a symmetric matrix by a random orthogonal projection

Given fixed real symmetric $D\in\mathbb{R}^{n\times n}$ with $n$ distinct eigenvalues, let $U$ be a random orthogonal matrix selected uniformly from the space of $n\times n$ orthogonal matrices, and ...
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Are the coefficients of a linear combination of random vectors as random?

Given are $2n$ random vectors $x_i,y_i\in\mathbb{C}^n$ for $i=1,\ldots,n$ which entries are drawn iid from some absolutely continuous distribution. Every set of $n$ different of those vectors is ...
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Solving a system of quadratic equations over a subspace

Let $A \subseteq \mathbb{R}^{n}$ be subspace of dimension $d$, parametrized as $A=\{x|Vx=0\}$, where $V$ is a suitable $d \times n$ matrix. Now a system of $m$ quadratic equations ...
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Positive solutions to simultaneous real quadratic equations

I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as $diag(x)Ax=1$ $x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal ...
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Complex conjugate and unitary complex conjugate

Definition: Let V be complex finite dimensional inner product space Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$ Definition: Anti-unitary Complex ...
Suppose we have an exterior algebra over $\mathbb{F_2}$, say $R = \Lambda_{\mathbb{F_2}}V$, where $V$ is an $n$-dimensional $\mathbb{F}_2$ vectorspace. Let $x_1,\ldots,x_n$ be a basis of that ...