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
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Linear Algebra, Eigenvectors, Vectors, Scalar, pg, Eigenvectors of Matrices

Click to edit the document details