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# 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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### How is the reduced matrix of a matrix?

I'm trying to generate the characteristic polynomial of the matrix below, from the reduced matrix. But the calculations are not correct, because, probably, my reduced matrix is wrong. I hope this ...
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### Square root of PSD matrix and unitary freedom

Consider a given PSD matrix $S$, which can be written as: $$S= A A^\top$$ where $A$ is its unique PSD square root. I am computing $A$ through the EVD of $S= U DU^\top$, i.e., $A= U D^{1/2}U^\top$. I ...
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### QR decompositon and Q is upper triangular

Suppose $A\in\mathbb{R}^{n\times n}$ and $D=\text{diag}\{d_1,\cdots, d_n\}\in\mathbb{R}^{n\times n}$. Construct an orthogonal $Q$ such that $$Q^T A-DQ^T=R$$ is upper triangular. I want to construct $Q$...
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### Best inverse / minimization solution of ill conditioned matrix and underdetermined system

I have an ill-conditioned matrix G representing Green's function estimates of a strain model for a rock mass that has been instrumented with optical fiber measuring strain. My problem is seemingly the ...
For example, suppose you have a known $\mathbf{A}$ and a known $\mathbf{Y}$ with an unknown $\mathbb{X}$: $$\mathbf{A}\mathbb{X} = \mathbf{Y}\\ \mathbb{X} = \mathbf{A^{-1}Y}$$ Now, suppose you ...