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From 6,
where and are the standard basis vectors for . We consider the case where ; the case where is
similar. Referring to Figure 4.3.5b, we have , so
and referring to Figure 4.3.5c, we have , so
Thus the standard matrix for T is
Solution (b) and , it follows from part (a) that the standard matrix for this projection operator is
Since
Thus
or, in point notation,
Geometric Interpretation of Eigenvectors
Recall from Section 2.3 that if A is an matrix, then is called an eigenvalue of A if there is a nonzero vector x such that
The nonzero vectors x satisfying this equation are called the eigenvectors of A corresponding to .
Eigenvalues and eigenvectors can also be defined for linear operators on ; the definitions parallel those for matrices.
DEFINITION
If is a linear operator, then a scalar is called an eigenvalue of T if there is a nonzero x in such that (7)
Those nonzero vectors x that satisfy this equation are called the eigenvectors of T corresponding to .
