Oxford University Mathematical Institute
Linear Algebra II for Mathematical Moderations
Lecture 5 (Tuesday 1 February 2011):
Eigenvalues and eigenvectors
The theory of eigenvalues and eigenvectors (to be defined shortly) applies equally
to square matrices and to endomorphisms of a finite-dimensional vector space V.
But we will, in fact, concentrate on matrices in the first instance.
Recall that the following three properties of an n×nmatrix Aare equivalent:
(1) Ais singular (in the sense that detA= 0 );
(2) Ais not invertible;
(3) there exists a non-zero column vector v= (x1, x2,...,xn)tr such that A v = 0.
Now we come to the main subject of the next few lectures.
Definition. Let A∈Mn×n(R). The scalar λ∈Ris said to be an eigenvalue
of Aif there exists a non-zero column vector v= (x1, x2,...,xn)tr ∈Rnsuch that
Av =λ v . The non-zero column vector v= (x1, x2,...,xn)tr ∈Rnis said to be
an eigenvector of Aif there exists a scalar λ∈Rsuch that A v =λ v .
Similarly, for an endomorphism T:V→Vof a finite-dimensional real vector
space, the scalar λ∈Ris said to be an eigenvalue of Tif there exists v∈V\ {0}
such that T v =λ v ; and the non-zero vector v∈Vis said to be an eigenvector of
Tif there exists a scalar λ∈Rsuch that T v =λ v .
Examples.
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