# Questions tagged [matrix-decomposition]

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

105 questions
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### Problem with Singular Value Decomposition

I have a very trivial SVD Example, but I'm not sure what's going wrong. The typical way to get an SVD for a matrix $A = UDV^T$ is to compute the eigenvectors of $A^TA$ and $AA^T$. The eigenvectors of ...
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### Polar decomposition of real matrices

Can we decompose real matrices in a way that is analogous to polar decomposition in the complex case? What I mean is: given an invertible real matrix $M$, can we always write: $$M = OP,$$ maybe ...
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### How do I find upper triangular form of a given 3 by 3 matrix??

We are asked to find an invertible matrix $P$ and an upper triangular matrix $U$ such that: $P^{-1}\begin{pmatrix} 3 & -1 & 1 \\ 2 & 0 & 0 \\ -1 & 1 & 3 \end{pmatrix}P=U$ I'm ...
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### Why isn't the SVD simplified to USV (no transpose)?

I've looked at a lot of explanations of the SVD, and they all phrase it as $A = USV^T$ (or $V^*$ for complex matrices), and where V is unitary. But the conjugate of a unitary matrix is always unitary, ...
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### How to prove whether a matrix has rank $1$

If $u$ $∈ \mathbb R^{m \times 1}$ and $v ∈ \mathbb R^{n \times 1}$ how do you show that the $(m \times n)$ matrix $uv^T$ has rank $1$? Would providing an example be sufficient to prove it?
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### Decompose a 2D arbitrary transform into only scaling and rotation

Related to: Newly Developed With Details - Describing orthographic projection using simple 2D transformations Given an arbitrary 2D linear transform (which may include shear, i.e. the vectors or the ...
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