# Questions tagged [matrix-decomposition]

Questions about matrix decompositions, such as the LU, Cholesky, SVD (Singular value decomposition) and eigenvalue-eigenvector decomposition.

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### Solving for $X$ using the SVD of $QX$ when $Q$ is orthogonal

I inherited some code (see below), and I am not quite sure what it does. It is part of a factor analysis-type model that learns a latent variable $X \in \mathbb{R}^{N \times K}$ with $N > K$ that ...
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### Why does SVD solve $\underset{U,V}{\min}\| A - UV^T\|_F^2$

I read here the following: You can solve the quadratic problem below through Singular Value Decomposition (SVD) of the matrix $A$. \begin{align} \underset{U,V}{\min} \| A - UV^T\|_F^2 \end{...
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### Solving $max \sum_{i=1}^2 \sum_{j=1}^2 (\phi_i A_{ij} \psi_j)^2$

Could you please provide some hints for me to solve this optimization problem? Here, for any $i=1,2$ and $j=1,2$, $\phi_i$ and $\psi_j$ are unknown vectors, $\alpha_i$, $\beta_j$ are some known ...
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### Eigenvectors of Matrix A [closed]

For a matrix Q prove the eigenvectors of kI-Q are equal to that of Q
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### Take derivative of matrix

A part of an objective function is: $$F=\|H-\mu_H\|_F^2$$ And we have: $$\mu_H=\frac{\Sigma H}{n_H}$$ In fact, $\mu_H$ is the average of $H$ in one dimension and is repeated $n$ times in which all ...
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### Square root of the product of a diagonal and symmetric matrix

If I have a diagonal matrix $D$ and a positive-definite symmetric matrix $C$, is there a formula for the square root of the product, $(DC)^{1/2}$? Also, $DC \neq CD$. What I have so far is \begin{...
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### Determinant of Square Root of Positive Define Matrix

Suppose the matrix $A \in \mathbb{R}^{n\times n}$ is positive definite symmetric. To begin, I want to investigate if the following equality holds $$|\det A^{1/2}| = |\det A|^{1/2}.$$ Since $A$ is ...