MATH348 - Advanced Engineering Mathematics 1
E. Kreyszig, Advanced Engineering Mathematics, 9th ed. Section 8.1, pgs. 334-339
Lecture: Eigenvalues and Eigenvectors Module: 07
Suggested Problem Set: {3, 5, 13, 14,16, 19, 21}June 25, 2009
Quote of Lecture 7
George Carlin: By and large, language is a tool for concealing the truth.
May 12, 1937 June 22, 2008
Okay, we know about Ax =b, or if we don’t then we have some places to look. Now we concentrate on
a special version of this equation where b=λx, λ ∈Cand we say that,
Ax =λx,(1)
is an eigenvalue-eigenvector problem for the square matrix An×n. Specifically, xis called the eigenvector
corresponding to the eigenvalue λ. If we think of Aas a linear transformation then λis a measure of
the transformation in the x-direction. The set of all eigenvectors and their corresponding eigenvalues then
provides yet another characterization of the transformation defined by A.
Solving (1) is a two part process:
•Calculate the characteristic equation from,
det(A−λI) = 0,
and find λby solving for the roots of the polynomial. These roots are often called the spectrum of A
and can be denoted at σ(A).
•Determine a basis for the null space of,
(A−λI),
by solving (A−λI)x=0. The basis vectors are eigenvectors associated with the particular λused to
calculate them. Sometimes, the collection of all eigenvectors is called an eigenbasis for A. I
If the eigenbasis of a matrix forms a basis for Rnthen many interesting properties can be deduced. If this
occurs and the matrix is self-adjoint, Ah=A, then one can show that the spectrum is purely real and that
the eigenbasis forms an orthonormal basis for Rn.1
Lecture Goals
•Understand how the concept of linear rescaling is related to eigenvalue-eigenvector problems.
•Use previous concepts of linear algebra to deduce a method for calculating eigenvalues and eigenvectors.
Lecture Objectives
•Derive auxiliary equations needed to calculate eigenvalues and eigenvectors.
•Summarize 2 ×2 theory.
•Calculate the eigenbasis of various ‘instructive’ matrices.
1This concept underpins the theoretical measurements of quantum particles and will be important in the study of
physical PDE.