The rank of a matrix

8.11 THE RANK OF A MATRIX
8.11 The rank of a matrix
The rank of a general M × N matrix is an important concept, particularly in
the solution of sets of simultaneous linear equations, to be discussed in the next
section, and we now discuss it in some detail. Like the trace and determinant,
the rank of matrix A is a single number (or algebraic expression) that depends
on the elements of A. Unlike the trace and determinant, however, the rank of a
matrix can be defined even when A is not square. As we shall see, there are two
equivalent definitions of the rank of a general matrix.
Firstly, the rank of a matrix may be defined in terms of the linear independence
of vectors. Suppose that the columns of an M × N matrix are interpreted as
the components in a given basis of N (M-component) vectors v1 , v2 , . . . , vN , as
follows:


↑ ↑
↑
A =  v1 v2 . . . vN  .
↓ ↓
↓
Then the rank of A, denoted by rank A or by R(A), is defined as the number
of linearly independent vectors in the set v1 , v2 , . . . , vN , and equals the dimension
of the vector space spanned by those vectors. Alternatively, we may consider the
rows of A to contain the components in a given basis of the M (N-component)
vectors w1 , w2 , . . . , wM as follows:


← w1 →
 ← w2 → 


A=
.
..


.
← wM
→
It may then be shown§ that the rank of A is also equal to the number of
linearly independent vectors in the set w1 , w2 , . . . , wM . From this definition it is
should be clear that the rank of A is unaffected by the exchange of two rows
(or two columns) or by the multiplication of a row (or column) by a constant.
Furthermore, suppose that a constant multiple of one row (column) is added to
another row (column): for example, we might replace the row wi by wi + cwj .
This also has no effect on the number of linearly independent rows and so leaves
the rank of A unchanged. We may use these properties to evaluate the rank of a
given matrix.
A second (equivalent) definition of the rank of a matrix may be given and uses
the concept of submatrices. A submatrix of A is any matrix that can be formed
from the elements of A by ignoring one, or more than one, row or column. It
§
For a fuller discussion, see, for example, C. D. Cantrell, Modern Mathematical Methods for Physicists
and Engineers (Cambridge: Cambridge University Press, 2000), chapter 6.
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