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The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix.
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Therefore, any permutation matrix P factors as a product of row-interchanging elementary matrices, each having determinant −1.
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Every real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a polar decomposition.
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Using the Jordan normal form, one can prove that every square real matrix can be written as a product of two real symmetric matrices, and every square complex matrix can be written as a product of two complex symmetric matrices.
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Any n × n permutation matrix can be constructed as a product of no more than n − 1 transpositions.
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The sum over all paths is given by a product of matrices, each matrix element is the transition probability from one slice to the next.
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The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose.
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Any orthogonal matrix of size n × n can be constructed as a product of at most n such reflections.
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Thus, for example, the product of a 1×n matrix and an n×1 matrix, which is formally a 1×1 matrix, is often said to be a scalar.
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Any rotation matrix of size n × n can be constructed as a product of at most n(n − 1)/2 such rotations.
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When K is the field of complex numbers, this shows that the algebra K is a product of several copies of complex matrix algebras, one for each irreducible representation.
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He also held minor parts in Australia-based international film productions including The Coca-Cola Kid, Quigley Down Under and The Matrix.
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Techniques such as casting, stamping and moulding use an intermediate matrix containing the design to produce the work; many of these allow the production of several copies.
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This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible.
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