\(A = CMR\) Factorization

Factor a matrix as \(A = CMR\)1, where \(M = W^{-1}\) and \(R\) is the matrix of linearly independent rows of \(A\).
\(W\) is the intersection submatrix2 at the intersection of pivot columns and rows.

The \(A = CMR\) factorization expresses a matrix as:

\[ A = CMR, \qquad M = W^{-1} \]

where:

  • \(C\) is an \(m \times r\) matrix of linearly independent columns from \(A\)
  • \(R\) is an \(r \times n\) matrix of linearly independent rows from \(A\)
  • \(W\) is the \(r \times r\) invertible submatrix at the intersection3, and \(M = W^{-1}\) is its inverse
  • \(r = \operatorname{rank}(A)\)

Construction:

  1. Identify pivot positions in \(\operatorname{RREF}(A)\)
  2. Extract pivot columns from \(A\) to form \(C\)
  3. Extract pivot rows from \(A\) to form \(R\)
  4. \(W\) is the submatrix of \(A\) at the intersection of these rows/columns; then \(M = W^{-1}\)

Applications:

  • Expresses \(A\) in terms of its essential structure
  • Generalizes LU factorization4
  • Useful in numerical linear algebra
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Computes the rank-revealing \(A = CMR\) factorization of the matrix.
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