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2 votes

How to decompose a simple $3\times3$ shear transformation into a rotation, scale, and rotation

For this special case I describe a simple approach so I can avoid a general SVD solver. We can compute the SVD $A = U \Sigma V^T$ via eigen-analysis of $$ \begin{eqnarray} A^T A &=& V \Sigma^...
wcochran's user avatar
  • 790
1 vote
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Unitary matrix factorization with diagonal matrix implies eigenvalues

You can just check it directly. Let $u_i$ by the $i$th column of $U$ and $\sigma_i$ be the $i$th diagonal entry of $\Sigma$. We then have $$ AA^*u_i = (U\Sigma\Sigma^*U^*)u_i = U\Sigma^2e_i = U(\...
whpowell96's user avatar
  • 6,199
1 vote
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Proof of the Single Value Decomposition

The argument is fine, but it's missing some details. First, let's choose $\pmb{v}$ as the eigenvectors of $A^TA$. This is a free choice, and we know it will produce a set of orthonormal vectors. ...
Martin Argerami's user avatar

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