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→Example: the multiplication by zero is not "implicit". let's also mention the compact SVD here |
m →Example |
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\end{bmatrix}</math>
and get an equally valid singular value decomposition. As the matrix {{tmath|\mathbf M}} has rank 3, it has only 3 nonzero singular values. In taking the product {{tmath|\mathbf{U}\mathbf{\Sigma} \mathbf{V}^* }}, the final column of {{tmath|\mathbf U}} and the final two rows of {{tmath|\mathbf{V^*} }} are multiplied by zero, so have no effect on
The [[#Compact SVD|compact SVD]], {{tmath|1= \mathbf M = \mathbf{U}_r\mathbf{\Sigma}_r \mathbf{V}_r^* }}, eliminates these superfluous rows, columns, and singular values:
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