rank of a matrix
the dimension of the subspace spanned by the columns of viewed as elements of the -dimensional left vector space over .
the dimension of the subspace spanned by the rows of viewed as elements of the -dimensional right vector space over .
the dimension of the subspace spanned by the rows of viewed as elements of the -dimensional left vector space over .
The numbers are respectively called the right column rank, left column rank, right row rank, and left row rank of , and they are respectively denoted by , , , and .
Since the columns of are the rows of its transpose , we have
In addition, it can be shown that for a given matrix ,
For any , it is also easy to see that the left column and row ranks of are the same as those of . Similarly, the right column and row ranks of are the same as those of .
If is a field, , so that all four numbers are the same, and we simply call this number the rank of , denoted by .
However, the left column rank is not necessarily the same as the right row rank of a matrix, if the underlying division ring is not commutative, as can be shown in the following example: let and be vectors over the Hamiltonian quaternions . They are columns in the matrix
Since , they are left linearly dependent, and therefore the left column rank of is . Now, suppose , with . Since , then , which boils down to two equations and , and which imply that , showing that are right linearly independent. Thus the right column rank of is .
|Title||rank of a matrix|
|Date of creation||2013-03-22 19:22:42|
|Last modified on||2013-03-22 19:22:42|
|Last modified by||CWoo (3771)|
|Defines||left row rank|
|Defines||left column rank|
|Defines||right row rank|
|Defines||right column rank|