Column-Row (CR) Factorization

Factor a matrix as \(A = CR\)1 using its column and row spaces.
\(C\) contains independent columns of \(A\), \(R\) contains rows of RREF\((A)\).2

The Column-Row (CR) factorization expresses a matrix as:

\[ A = CR \]

where:

  • \(C\) is an \(m \times r\) matrix containing the linearly independent columns of \(A\)
  • \(R\) is an \(r \times n\) matrix containing the nonzero rows of RREF\((A)\)
  • \(r = \text{rank}(A)\)

How it works:

  1. Compute RREF\((A)\) to identify pivot columns3
  2. \(C\) = columns of \(A\) corresponding to pivot positions
  3. \(R\) = nonzero rows of RREF\((A)\)

Applications:

  • Reveals the rank and basis for column space
  • Efficient storage for low-rank matrices
  • Understanding matrix structure and dependencies
Set matrix dimensions

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

Display values as:
Set Up
Loads a pre-filled 3×4 matrix — ready to compute.
Fills the matrix with random integer entries.
Compute
Computes the rank-revealing A = CR factorization using pivot columns and RREF rows.
Resets all dimensions, matrix entries, and results.