8.11 THE RANK OF A MATRIX
8.11 The rank of a matrixTherankof a generalM×Nmatrix 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 matrixAis a single number (or algebraic expression) that depends
on the elements ofA. Unlike the trace and determinant, however, the rank of a
matrix can be defined even whenAis not square. As we shall see, there are two
equivalentdefinitions of the rank of a general matrix.
Firstly, the rank of a matrix may be defined in terms of thelinear independenceof vectors. Suppose that the columns of anM×Nmatrix are interpreted as
the components in a given basis ofN(M-component) vectorsv 1 ,v 2 ,...,vN,as
follows:
A=↑↑ ↑
v 1 v 2 ... vN
↓↓ ↓.Then therankofA, denoted by rankAor byR(A), is defined as the number
oflinearly independentvectors in the setv 1 ,v 2 ,...,vN, and equals the dimension
of the vector space spanned by those vectors. Alternatively, we may consider the
rows ofAto contain the components in a given basis of theM(N-component)
vectorsw 1 ,w 2 ,...,wMas follows:
A=
← w 1 →
← w 2 →
..
.
← wM →
.It may then be shown§ that the rank ofAis also equal to the number of
linearly independent vectors in the setw 1 ,w 2 ,...,wM. From this definition it is
should be clear that the rank ofAis 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 rowwibywi+cwj.
This also has no effect on the number of linearly independent rows and so leaves
the rank ofAunchanged. 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 usesthe concept ofsubmatrices. A submatrix ofAis any matrix that can be formed
from the elements ofAby 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.