# Tagged Questions

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### Maximizing the trace

Say i have the following maximization. $max_R$ trace $(RZ): R^TR = I_n$ where $R$ is an $n$ x $n$ orthogonal transformational vector. Also, the SVD of $Z = USV^T$. I'm trying to find the optimal ...
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### Matrix Partial Derivative?? NMF Multiplicative update rules

Recently, I read Lee & Seung's work on Nonnegative Matrix Factorization. But I have problem with the update rule: The object function is minimize: $\|V - MH \|$ with respect to M and H, subject ...
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### Quadratic minimization with real output

Let $A$ be a $n\times n$ hermitian matrix. My goal is to minimize the following hermitian form with an additionnal "real constraint": $$\min_f f^\ast A f$$ $$\text{subject to }f \in \mathbb{R}^n$$ ...
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### Expanding variance

Could someone please expand on line 2 and 3 of: Thank you.
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### Minimum of a quadratic form

If $\bf{A}$ is a real symmetric matrix, we know that it has orthogonal eigenvectors. Now, say we want to find a unit vector $\bf{n}$ that minimizes the form: $${\bf{n}}^T{\bf{A}}\ {\bf{n}}$$ How can ...
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### Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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### steepest descent with quadratic form converge in 1 iteration

Well I'm stuck on an exercise given: The steepest descent method is applied to the quadratic form $$Q(\mathbf{x}) = \tfrac{1}{2}\mathbf{x}^TA\mathbf{x} - \mathbf{b}^T\mathbf{x} + c$$ where $A$, ...
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### Direction of steepest descent and minimization?

I have the following linear function: $min$ 1/2 $<x, x>$ + $r^Tx$ for every x belonging to $R^n$, $r^Tx$ belongs to $R^n$ Now, = $x^TAy$ and A is symmetric positive definite. = $x^TAy$ is ...
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### What (if anything) can I say about the inverse of the matrix product B'AB if B is not square?

Suppose I have: a matrix $A$ with dimension $n \times n$ a matrix $B$ with dimension $n \times m$. $C = B^{T}AB$. I'm interested in finding an expression for $C^{-1}$ when $m < n$. The ...
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### Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
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### find conditions on input data such that a linear system has (no) feasible points

As a result of the apllication of Farkas' lemma I obtained the following problem: Let $m,n,q \in \mathbb{N}$, $b \in \mathbb{N}^m, l \in \mathbb{N}^m$ with $l_i \mid q$ for all $i=1,\ldots,m$. ...
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### Monotononically Increasing Water Filling Solution?

$\mathbf{I}$ is the $K\times K$ identity matrix. $\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors. Consider the solution of the convex optimisation problem over ...
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### Optimization - show that linearized feasible set is empty.

I need help in the following problem: Consider the following optimization problem $$\min_{x_1,x_2}-x_1-x_2\quad\text{s.t.}\quad x_1^2+x_ 2^2-1=0,\quad x_1,x_2\geqslant 0.$$ Show that the ...
I'm working on an optimization problem and am stuck at this particular step. Let $\bf{A}$ be a matrix with 4 columns and a finite number of rows, consisting of elements which are either 0 or 1. Let ...
The classical orthogonal Procrustes problem concerns finding the matrix $\Omega$ which minimizes $||A\Omega-B||_{F}$ subject to $\Omega'\Omega=I$, with A and B known matrices. Let A be the identity. I ...