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Transcript
Defn: A set V together with two operations, called addition
and scalar multiplication is a vector space if the following
vector space axioms are satisfied for all vectors u, v, and w
in V and all scalars, c, d in R.
Vector space axioms:
a.) u + v is in V
b.) cu is in V
c.) u + v = v + u
d.) (u + v) + w = u + (v + w)
e.) There is a vector, denoted by 0, in V such that
u + 0 = u for all u in V
f.) For each u in V , there is an element, denoted by
−u, in V such that u + (−u) = 0
g.) (cd)u = c(d u)
h.) (c + d )u = cu + d u
i.) c(u + v) = cu + cv
j.) 1u = u
Examples:
1.) Rk with the usual operations of addition and scalar multiplication is a vector space.
2.) The set M k,n , the set of all k × n matrices with the usual
operations of addition and scalar multiplication is a vector
Linear Algebra Review: Eigenvalues and Eigenvectors
Defn: λ is an eigenvalue of the linear transformation
T : V → V if there exists a nonzero vector x in V such
that T (x) = λx. The vector x is said to be an eigenvector
corresponding to the eigenvalue λ.
4 1
Example: Let T (x) =
x.
5 0
−1
1
−1
Note
=
= −1
5
−5
5
−1
is a corresponding
Thus -1 is an eigenvalue of A and
5
eigenvector of A.
Note
4 1
5 0
4 1
5 0
1
5
1
=
=5
1
5
1
1
Thus 5 is an eigenvalue of A and
is a corresponding
1
eigenvector of A.
4 1
2
16
2
Note
=
6= k
for any k.
5 0
8
10
8
2
Thus
is NOT an eigenvector of A.
8
MOTIVATION:
2
−1
1
Note
=
+3
8
5
1
1
−1
1
−1
2
+ 3A
)=A
Thus A
= A(
+3
1
5
8
5
1
−1
1
16
= −1
+3·5
=
5
1
10
Thus to find the eigenvalues of A and their corresponding
eigenvectors:
Step 1: Find eigenvalues: Solve the equation
det(λI − A) = 0 for λ.
Step 2: For each eigenvalue λ0 , find its corresponding eigenvectors by solving the homogeneous system of equations
(λ0 I − A)x = 0 for x.
Defn: det(λI − A) = 0 is the characteristic equation of A.
Finding eigenvalues:
Suppose Ax = λx
(Note A is a SQUARE matrix).
Then Ax = λIx where I is the identity matrix.
Thm 3: The eigenvalues of an upper triangular or lower triangular matrix (including diagonal matrices) are identical to
its diagonal entries.
Defn: The eigenspace corresponding to an eigenvalue λ0 of
a matrix A is the set of all solutions of (λ0 I − A)x = 0.
Thus λIx − Ax = (λI − A)x = 0
Thus if Ax = λx for a nonzero x, then (λI − A)x = 0 has a
nonzero solution.
Thus det(λI − A)x = 0.
Note that the eigenvectors corresponding to λ are the nonzero
solutions of (λI − A)x = 0.
Note: An eigenspace is a vector space
The vector 0 is always in the eigenspace.
The vector 0 is never an eigenvector.
The number 0 can be an eigenvalue.
Thm: A square matrix is invertible if and only if λ = 0 is not
an eigenvalue of A.