# 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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### Sufficient/necessary condition for submatrix determinant (minor) that decreases with size?

Definition. Given a square matrix ${\bf{A}}=[a_{ij}] \in {\mathbb{C}^{n \times n}}$, the submatrix ${{\bf{A}}_{{i_1},{i_2},...{i_k}}}$ is formed by retaining the $({i_1},{i_2},...{i_k})$-th rows and ...
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### Show that partial pivoting leads to an $LU$ decomposition of $PA$.

Exercise: show that for every non-singular matrix $A$, partial pivoting leads to an $LU$ decomposition of $PA$ so: $PA = LU$. I have the following theorems I can use: Theorem 1: Assume that the ...
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### Is a principal submatrix of a diagonalizable matrix diagonalizable?

Let $A$ be diagonalizable, i.e., $A=X \Lambda X^{-1}$ for some diagonal matrix $\Lambda$. Consider $B$ which is a principal submatrix of $A$. Does there exist an invertible matrix $Y$ and a diagonal ...
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### PCA derivation by minimizing reconstrution squared error plus orthonormality constraints

I'm trying to better understand the link between PCA and Matrix Factorization of the form $X \approx WH$. I've read somewhere that the PCA solution can be also derived from the following cost ...
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