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It follows from part (a) that the formula for T in matrix notation is
Substituting in this formula yields
so
Thus .
Eigenvalues of a Linear Operator
Eigenvectors and eigenvalues can be defined for linear operators as well as matrices. A scalar λ is called an eigenvalue of a
linear operator if there is a nonzero vector x in V such that . The vector x is called an eigenvector of T
corresponding to λ. Equivalently, the eigenvectors of T corresponding to λ are the nonzero vectors in the kernel of
(Exercise 15). This kernel is called the eigenspace of T corresponding to λ.
EXAMPLE 4 Eigenvalues of a Linear Operator
Let and consider the linear operator T on V that maps to . If , then
, so is an eigenvector of T associated with the eigenvalue 1:
Other eigenvectors of T associated with the eigenvalue 1 include , , and the constant function 3.
It can be shown that if V is a finite-dimensional vector space, and B is any basis for V, then
1. The eigenvalues of T are the same as the eigenvalues of .
2. A vector x is an eigenvector of T corresponding to λ if and only if its coordinate matrix is an eigenvector of
corresponding to λ.
We omit the proofs.
EXAMPLE 5 Eigenvalues and Bases for Eigenspaces
