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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
.
f.
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 Spring '08
 Staff
 Eigenvectors, Vectors, Scalar

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