# What is matrix rank and how do i calculate it?

The rank of a matrix \(A\) is the number of independent columns of \(A\). A square matrix is full rank if all of its columns are independent. That is, a full rank matrix has no column vector \(v_i\) of \(A\) that can be expressed as a linear combination of the other column vectors \(v_j \neq \Sigma_{i = 0, i\neq j}^{n} a_i v_i\). For example, if one column of \(A\) is twice another one, then those two columns are linearly dependent (with a scaling factor 2) and thus the matrix would not be full rank.

A simple test for determining if a matrix is full rank is to calculate its determinant. If the determinant is zero, there are linearly dependent columns and the matrix is not full rank. Prof. John Doyle also mentioned during lecture that one can perform the singular value decomposition of a matrix, and if the lowest singular value is near or equal to zero the matrix is likely to be not full rank ("singular").

--Sawyer Fuller 16:22, 29 October 2007 (PDT)