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Informally, an algorithm is a set of explicit instructions used to solve a problem (e.g. Euclid's algorithm for computing the greatest common divisor of two integers). For more specific questions on algorithms, this tag may be used in conjunction with the approximation-algorithms, algorithmic-randomness and algorithmic-topology tags.
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Simultaneous Smith Normalization of a Composable Matrix Sequence
The Smith Normal Form of a product of matrices is not, in general, the product of the Smith Normal Forms. So most of the time you can't do this.
There are some results in this direction in the book …