Last updated October 14, 2026 · Report an error
Rank factorization expresses a matrix A as the product of two smaller matrices, C and F, whose shared inner dimension is exactly A's rank — a compact way to represent a matrix using only its independent information.
This calculator finds the rank factorization A = CF for any matrix — square or rectangular — with rows and columns each from 2 up to 6.
How it works
Enter the number of rows and columns and the matrix values — they don't need to match. The calculator reduces the matrix to reduced row echelon form to find the rank and pivot columns, builds C from the original matrix's pivot columns, and builds F from the non-zero rows of the row echelon form.
- Click Calculate to see your results.
Examples
Rank-1 example
The matrix [[1,2],[2,4]] has rank 1, factoring into C = [[1],[2]] and F = [[1,2]] — multiplying C by F reconstructs the original matrix exactly.
Who should use it
- Checking linear algebra coursework on rank factorization and matrix decomposition.
- Understanding the relationship between a matrix's rank, column space, and row echelon form.
Industry applications
- Linear algebra education
- Data compression and dimensionality reduction (rank factorization underlies many low-rank approximation techniques)
Advantages
- Works for both full-rank and rank-deficient (singular) matrices, square or rectangular.
- Uses genuine columns and rows from the original matrix and its row echelon form, not an abstract factorization.
Limitations
- Capped at 6 rows and 6 columns to keep computation instant on a public, synchronous endpoint.
Common mistakes to avoid
- Expecting C and F to have the same dimensions as A — their shared inner dimension is the rank, which is often smaller than the full matrix size.
- Forgetting to check that C × F actually reconstructs A as a sanity check on the result.
Best practices
- Multiply the returned C and F matrices back together to confirm they reconstruct the original matrix A.
Tips
- Rank factorization is the conceptual foundation behind low-rank matrix approximation techniques used in data compression and machine learning.