In mathematics, given a field, and a matrix, a rank decomposition or rank factorization of is a factorization of of the form is the rank.

Existence

Every finite-dimensional matrix has a rank decomposition: matrix whose column rank linearly independent; equivalently, the dimension of the column space be any basis is a linear combination-th column, then

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Non-uniqueness

of compatible dimensions.

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Construction

Rank factorization from reduced row echelon forms

, the reduced row echelon form all non-pivot columns.

Note: For a full-rank square matrix identity matrix).

Example

Consider the matrix

is in reduced echelon form.

, so

It is straightforward to check that

Proof

permutation matrix in block partitioned.

which is a product of elementary matrices, and the proof is complete.

Singular value decomposition

via a singular value decomposition

.

Consequences

rank(A) = rank(AT)

, the column rank equals its row rank.

Proof:. From the definition of matrix multiplication is a linear combination.

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.

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Notes

References