Magic Factorization

Factor a matrix as \(A = CW^{-1}R^*\)1 using independent columns and rows.
\(W\) is the "magic" submatrix2 at the intersection of pivot columns and rows.

The magic factorization expresses a matrix as:

\[ A = CW^{-1}R^* \]

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
  • \(r = \operatorname{rank}(A)\)

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 intersection of these rows/columns

Applications:

  • Expresses \(A\) in terms of its essential structure
  • Generalizes LU factorization4
  • Useful in numerical linear algebra
Set matrix dimensions

Enter dimensions and click Generate Matrix to create the input grid.

Display values as:
1 — Set Up
Loads a pre-filled 3×4 matrix — ready to compute.
2 — Compute
Computes the rank-revealing magic factorization of the matrix.
Resets all dimensions, matrix entries, and results.