Editing Rank (linear algebra) (section)

==Computing the rank of a matrix==

=== Rank from row echelon forms ===
{{main|Gaussian elimination}}
A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally [[row echelon form]], by [[elementary row operations]]. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of [[Pivot_element|pivots]] (or basic columns) and also the number of non-zero rows.

For example, the matrix {{mvar|A}} given by
<math display="block">A=\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}</math>
can be put in reduced row-echelon form by using the following elementary row operations:
<math display="block">\begin{align}
\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}
&\xrightarrow{2R_1 + R_2 \to R_2}
\begin{bmatrix}1&2&1\\0&1&3\\3&5&0\end{bmatrix}
\xrightarrow{-3R_1 + R_3 \to R_3}
\begin{bmatrix}1&2&1\\0&1&3\\0&-1&-3\end{bmatrix} \\
&\xrightarrow{R_2 + R_3 \to R_3}
\,\,
\begin{bmatrix}1&2&1\\0&1&3\\0&0&0\end{bmatrix}
\xrightarrow{-2R_2 + R_1 \to R_1}
\begin{bmatrix}1&0&-5\\0&1&3\\0&0&0\end{bmatrix}~.
\end{align}</math>
The final matrix (in reduced row echelon form) has two non-zero rows and thus the rank of matrix {{mvar|A}} is 2.

=== Computation ===
When applied to [[floating point]] computations on computers, basic Gaussian elimination ([[LU decomposition]]) can be unreliable, and a rank-revealing decomposition should be used instead. An effective alternative is the [[singular value decomposition]] (SVD), but there are other less computationally expensive choices, such as [[QR decomposition]] with pivoting (so-called [[rank-revealing QR factorization]]), which are still more numerically robust than Gaussian elimination. Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application.