# Tagged Questions

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### Max and Min using Lagrange Multipliers

Suppose A is a symmetric matrix. Show that the maximum and minimum of $\mathbf x ^T A \mathbf x$ subject to the constraint $\mathbf x ^T \mathbf x=1$ are the maximum and minimum eigenvalues of A. I ...
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### Symmetric Positive Definite Matrix Proof

Suppose that $H^+ = H - (\mathbf y^TH \mathbf y)^{-1} H\mathbf y \mathbf y^T H + (\mathbf y ^T \mathbf s )^{-1}\mathbf s \mathbf s^T$ where H is symmetric and positive definite. Supposing that ...
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### How to calc $\min ||J\Delta\tau + D||_*$

How to calculate $$\min_{\tau} ||J_1 \tau_1 + \cdots + J_p \tau_p + D ||_*$$ where $\tau_1, \cdots, \tau_p \in \mathbb{R}$ $J_1, \cdots, J_p, D \in \mathbb{R}^{m \times n}$ $||\cdot||_*$ is sum ...
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### Minimizing the following objective function with matrices

Suppose $A$ and $B$ are known matrices, and we are to find matrix $X$ that minimizes the following function, $$\frac{1}{2}||X||^2+\frac{1}{2}||X^TAX-B||^2$$ Taking the relevant derivative w.r.t $X$ ...
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### Minimize Frobenius norm with constraints

As a follow-up on my previous question, I would like to solve the following optimization problem: $\min \Vert MA-B \Vert_F^2-x^HMy\;\;s.t.\;\;M^HM=I$ where $A$ and $B$ are $N\times L$ complex ...
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### A simple optimization problem

$$f = x^Tx$$ $$g = Ax-b$$ The constraint is $Ax-b = 0$ I calculated $J' = f'+\lambda g'$ which is $2x^T+\lambda A^T = 0$ and $Ax-b=0$ . I dont know what to do next please help me out .
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### Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
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### Largest eigenvalue of symmetric matrix

I am trying to understand why the $\lambda_{\max}$ function is convex given an $n\,x\,n$ symmetric matrix, let's call it $A$. I know from elementary property of eigenvalues that all the eigenvalues of ...
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### What are non-orthogonal eigenvectors?

Given a symmetric matrix $A$, the maximum of the trace, $Tr(Z^TAZ)$ under the assumption that $Z^TZ=I$ occurs when $Z$ has the eigenvectors of $A$, as $Tr(U^TAU)= \lambda_1 +\lambda_2+...\lambda_ d$ ...
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### Solve: This System of equations for $X$ (does a real solution, exist?)
How can I solve $AX + diag(X)[I-c]=0$ for $X$? All matrices have real entries, $diag(X)$ is a diagonal matrix with the diagonal entries being the diagonal entries of $X$, and $c$ is a constant, real ...