Chapter 4 - Eigenvalues and Eigenvectors - Chapter 4...

Chapter 4 – Eigenvalues and Eigenvectors
Section 4.1 – Introduction to Eigenvalues and Eigenvectors
Pg 253, Definition – Let A be an matrix. A scalar λ is called an eigenvalue of A if there is a nonzero vector x such
that . Such a vector x is called an eigenvector of A corresponding to λ .
Pg 255, Definition – Let A be an matrix and let λ be an eigenvalue of A. The collection of all eigenvectors
corresponding to λ , together with the zero vector, is called the eigenspace of λ and is denoted by E
λ
.
Section 4.2 – Determinents
Pg 263, Definition – Let . Then the determinant of A is the scalar
Pg 264, Definition – Let be an matrix, where . Then the determinant of A is the scalar
Pg 265, Theorem 4.1 – The Laplace Expansion Theorem – The determinant of an matrix , where can be
computed as
(which is the cofactor expansion along the ith row ) and also as
(the cofactor expansion along the jth column ).
Pg 268, Theorem 4.2 – The determinant of a triangular matrix is the product of the entries on its main diagonal.
Specifically, if is an triangular matrix then
Pg 268, Theorem 4.3 – Properties of Determinants – Let be a square matrix.
a.
If A has a zero row (column), then .
b.
If B is obtained by interchanging two rows (columns) of A, then .
c.
If A has two identical rows (columns), .
d.
If B is obtained by multiplying a row (column) of A by k, then .
e.
If A, B, and C are identical except that the ith row (column) of C is the sum of the ith rows (columns) of A
and B, then .