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Home Math Background Matrices Rank of a Matrix
See also: matrix determinant, order of a matrix, linear dependence, PCA of Transposed Matrices 

Rank of a Matrix

The order of a matrix is a measure of its shape and size. However, the order does not provide any hints on the information content of a matrix. The 53-matrices

[ 1
0
0
0
0		 0
0
0
0
0		 0
0
0
0
0		 ]
[ 1
-2
7
3
-2		 0
3
0
-6
4		 -2
4
-14
-6
4		 ]
[ -2
7
1
5
7		 -1
-3
7
-1
4		 4
-3
-4
-4
7		 ]

differ in their information content, since the first and the second matrix contain rows and columns which are multiples of other rows and columns (some rows/columns are linearly dependent). The concept of linear independence leads to the definition of the row and column rank of an arbitrary matrix A:

Row Rank
Column Rank		 The maximum number of linearly independent rows in A is called the row rank of A; the maximum number of linearly independent columns in A is called the column rank of A.

It is a very important, and somewhat even surprising, result of matrix theory that row and column rank of a given matrix are always equal, no matter how the matrix is shaped. Thus, we don't have to distinguish between row and column rank of a matrix - we simply speak of the rank of a matrix.
 


Home Math Background Matrices Rank of a Matrix