Matrix Rank Calculator
Count the pivots surviving Gaussian elimination with partial pivoting and a 1e-10 tolerance.
Description
Estimate the rank of a rectangular matrix by counting pivots during Gaussian elimination.
Matrix rank is the number of linearly independent rows or columns. It reveals the dimension of the space represented by a matrix and helps identify redundant equations, singular square matrices, and feature collinearity.
When to use Matrix Rank Calculator
- Check whether rows or columns are linearly dependent
- Determine whether a square matrix has full rank
- Inspect redundancy in a small system or feature matrix
How the calculation works
Gaussian elimination with partial pivoting searches each column for a usable pivot, eliminates entries below it, and increments the rank for every accepted pivot. A numeric tolerance prevents tiny floating-point residue from being counted as an independent direction.
Interpreting the result
Rank cannot exceed the smaller of row count and column count. A square matrix is invertible only when its rank equals its dimension. Rank deficiency tells how many independent directions are missing, but not which data columns should be removed.
Important limitations
- Numeric rank depends on the 1e-10 pivot tolerance and the scale of the entries.
- Near-dependent rows can change classification after rescaling or measurement noise.
- Use domain-aware conditioning analysis for high-stakes numerical decisions.