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Ch 7.3: Sys. Lin. Eq., Lin. Ind., Eigenvalues^ •
A system of
n
linear equations in
n
variables,
2
, 2
2 (^2) , 2
1 (^1) , 2
1
, 1
2 (^2) , 1
1 (^1) , 1
n n
n n
b x a x a x a b x a x a x a =
L L
can be expressed as a matrix equation
Ax
=
b
: ,
,
2 (^2) ,
1 (^1) ,
n n nn
n
n^
b x a x a x
a^
+^
L
M
can
be expressed as a matrix equation
Ax
b :
n^ n
b b
x x
a
a
a
a
a
a
L L
1 2
1 2
2
(^22)
(^12)
, 1
(^2) , 1
(^1) , 1
⎜ ⎜ ⎜⎜⎝^
n
n nn
n
n
n
b
x
a
a
a
M
M
L
M
O
M
M
2
2
,
(^2) ,
(^1) ,
, 2
(^2) , 2
(^1) , 2
To see that the matrix equation is equivalent to the system of equations,
multiply the matrix and vector on the left side and equate components.
-^
If^
b^
=^
0 , the system is said to be
homogeneous
Nonsingular (i.e., Invertible) CaseNonsingular
(i.e., Invertible) Case
•^
An invertible matrix
A
is called nonsingular
An invertible matrix
A
is called nonsingular.
•^
If
A
is invertible, we can always solve
Ax
b
as follows:
b
A
x
b
A
Ix
b
A
Ax
A
b
Ax
−^
= •^
This solution is unique.
•^
If
A
is invertible, the only solution to
Ax
^0
is
the trivial solution
A
(^1)
x
A
^0
Example 1: Nonsingular Case
(2 of 2)
-^
Now let’s solve the nonhomogeneous linear system A
b
b l
i^
A
(^1)
A
x^
=^
b
below using
A
3
2
1
3
2 1
x
x
x
x
x x
-^
This system of equations can be written as
Ax
b
where
3
2
1
3
2
1
x
x x
x
x
x
where
⎞⎟ ⎟ ⎟ ⎠ ⎛⎜ −⎜ ⎜ ⎝ = ⎞⎟ ⎟ ⎟ ⎠ ⎛⎜ ⎜ ⎜ ⎝ = ⎞⎟ ⎟ ⎟ ⎠
⎛⎜ ⎜ ⎜ ⎝
2 2 0
,
, 8 3 4
3 0 1
2 1 0
1 2
b
x
A^
x x
-^
Then
⎞ ⎟ ⎟
⎛ −⎜ ⎜ ⎞⎟ ⎟ ⎛⎞⎜⎟ ⎜⎟
⎛^ ⎜ ⎜
−
−^
23
2 (^2) / 3 7 (^2) / 9
1
⎟ ⎠ ⎜ ⎝
⎟ ⎠ ⎜ ⎝
⎟ ⎠
⎜ ⎝^
−^
0
8 3 4
x^3
⎟ ⎟ ⎠ −=⎜ ⎜ ⎝ ⎟ ⎟ ⎠ −⎜⎟ ⎜⎟ ⎝⎠
⎜ ⎜ ⎝^
−
−
−
=
=^
−
12 7
2 0 (^2) / 1 2 (^2) / 3
1
4 2
1 b A x
Singular Case
-^
If the coefficient matrix
A
is singular (not invertible),
then either there is no solution to
Ax
b
, or there are
i fi it l
l ti
t^
A
b
infinitely many solutions to
A
x^
=^
b
We can find these solutions by row reducing
(A | b).
Example 3: Linear Independence
(1 of 2)
Example
3: Linear Independence
(1 of 2)
-^
Determine whether the following vectors are linearDetermine whether the following vectors are lineardependent or linearly independent.
⎞ ⎟ ⎛⎜
⎞⎟ ⎛⎜
⎞⎟ ⎛^ ⎜
2
1
0
) (^3) (
) (^2) (
) (^1) (
⎟ ⎟ ⎟⎠ ⎜ ⎜ ⎜⎝ =
⎟ ⎟ ⎟⎠ ⎜ =⎜ ⎜−⎝
⎟ ⎟ ⎟⎠ ⎜ =⎜ ⎜⎝
3 8
, 0 3
, (^1 )
) (^3) (
) (^2) (
) (^1) (
x
x
x
-^
We need to determine for what coefficients
x
x
x^
+^
) (^3) ( 3 ) (^2) ( 2 ) (^1) ( 1
c
c
c
or
⎞ ⎟ ⎟ ⎛⎜ =⎜ ⎞⎟ ⎟ ⎛⎞⎜⎟ ⎜⎟
⎛⎜ ⎜ ⇔ ⎞⎟ ⎟ ⎛⎜ =⎜ ⎞⎟ ⎟ ⎛⎜ ⎜ ⎞⎟ +⎟ ⎛⎜ ⎜ ⎞⎟ +⎟ ⎛⎜ ⎜
0 0
3 0 1
2 1 0
0 0
2 3
(^10)
0 1
c^1 c
c
c
c
⎟ ⎟⎠ =⎜ ⎜⎝ ⎟ ⎟⎠ ⎜⎟ ⎜⎟⎝⎠
⎜ ⎜⎝^
−
⇔ ⎟ ⎟⎠ =⎜ ⎜⎝ ⎟ ⎟⎠ ⎜ ⎜⎝ +⎟ ⎟⎠ ⎜ ⎜−⎝ +⎟ ⎟⎠ ⎜ ⎜⎝^
0 0
8 3 4
3 0 1
0 0
3 8
0 3
(^14)
2 3
3
2
1
c^ c
c
c
c
Example 3: Linear Independence
(2 of 2)
Example
3: Linear Independence
(2 of 2)
-^
We reduce the augmented matrix (
A
| b
We reduce the augmented matrix (
A
| b
(^
)^
⎞ ⎟ ⎟ ⎟
⎛⎜ ⎜ ⎜ ⎞⎟ →⎟ ⎟
⎛⎜ ⎜ ⎜ =^
0 2 1 0
0 3 0 1 0 3 0 1
0 2 1
0
b A (
)
⎞ ⎟ ⎟ ⎛⎜ ⎜
=
⎟ ⎟⎠
⎜ ⎜⎝ ⎟ ⎟⎠
⎜ ⎜⎝^
−
0 0
0
2
0
3
0 1 0 0 0 8 3 4
3
1
c
c
Th
th
l^
l ti
i^
d
⎟ ⎟⎠ ⎜ ⎜⎝ = →
= =
→
0 0
0 0
2
3 3
2
c
c c
c
-^
Thus, the only solution is
c
c
2
cn
= 0, and
therefore the original vectors are linearly independent.
Example 4: Linear Dependence
(2 of 2)
Example
4: Linear Dependence
(2 of 2)
-^
We thus reduce the augmented matrix (
A
| b
) as
We thus reduce the augmented matrix (
A
| b
), as
before.^ (
)^
⎞ ⎟ ⎟ ⎟
⎛^ ⎜ ⎜ ⎜
−
−
⎞⎟ →⎟ ⎟
⎛⎜ −⎜ ⎜
− −
=^
0 5 3
0
0 1 2 1 0 6 5 1
0 1 2 1
b A (^
)
⎞ ⎟ ⎟ ⎛⎜ ⎜
⎞⎟ ⎟
⎛ −⎜ ⎜
=
−
−
⎟⎠
⎜⎝ ⎟⎠
⎜⎝^
−
7 5
(^3) / 5
(^3) / 7
0
5
3
0
1
2
0 0 0 0 0 5 4 5
3
3
2
1
k
c
c
c
c
Thus the original vectors are linearly dependent with
⎟ ⎟⎠ ⎜ ⎜−⎝ = →⎟ ⎟⎠
−⎜ ⎜⎝
→ = =
→
5 3
(^3) / 5
0
0
0
5
3
3 3
3 3
2
k
c c
c c
c^
c
c
Thus
the original vectors are linearly dependent, with
⎞ ⎟ ⎟ ⎛⎜ ⎜ ⎞⎟ ⎟ ⎛ −⎜ ⎜ ⎞⎟ ⎟ ⎛ −⎜ ⎜ ⎞⎟ +⎟ ⎛⎜ ⎜
0 0
(^16) 3 2 5 5 1 1 7
⎟ ⎟⎠ ⎜ ⎜⎝ =⎟ ⎟⎠
⎜ ⎜⎝ −⎟ ⎟⎠
⎜ ⎜−⎝ +⎟ ⎟⎠ −⎜ ⎜⎝
(^00)
6 5 3 5 4 5 (^15) 7
Linear Independence and InvertibilityLinear
Independence and Invertibility
-^
Consider the previous two examples:
-^
Consider the previous two examples:– The first matrix was known to be nonsingular, and its
column vectors were linearly independent.
y^
p
- The second matrix was known to be singular, and its
column vectors were linearly dependent.
-^
This is true in general: the columns (or rows) of A
are linearly independent if and only if
A
is
nonsingular if and only if
A
exists
nonsingular if and only if
A
(^1)
exists.
-^
Also,
A
is nonsingular if and only if det
A
hence columns (or rows) of
A
are linearly
hence columns (or rows) of
A
are linearly
independent if and only if det
A
Eigenvalues and EigenvectorsEigenvalues
and Eigenvectors
-^
The eqn
Ax
=
y
can be viewed as a map of the
The eqn.
Ax
y
can be viewed as a map of the
vector
x
into a new vector
y
.
-^
Nonzero vectors
x
that
A
maps to multiples of
themselves are important in applications.
-^
We
will be interested in finding numbers
λ^
and
corresponding vectors
x
such that
Ax
=
λ
x
or
corresponding vectors
x
such
that
Ax
=
λ
x
or
equivalently, (
A
-^ λ
I )
x
=
^0
.
-^
This equation has a nonzero solution if we
q
choose
λ^
such that det (
A
-^ λ
I ) = 0.
-^
Such values of
λ^
are called
eigenvalues
of
A
,
and the corresponding nonzero solutions
x
are
and the corresponding nonzero solutions
x
are
called
eigenvectors
.
Example 5: Eigenvalues
(1 of 3)
Example
5: Eigenvalues
(1 of 3)
-^
Find the eigenvalues and eigenvectors of the matrix
A
Find the eigenvalues and eigenvectors of the matrix
A
=^
A
-^
Solution: Choose
λ^
such that det(
A
-^ λ
I ) = 0, as follows.
⎛^
(^
det
det
⎛^ ⎜
λ I
A
(^
)(^
)^
det
(^
)(^
2
Example 5: Second Eigenvector
(3 of 3)
Example
5: Second Eigenvector
(3 of 3)
-^
Eigenvector for
λ^
=^
- 7:
Solve
Eigenvector
for
λ^
Solve
(^
)^
−^
1 2
1 2
x x
x x
x I
A
by row reducing the augmented matrix:
⎝^
+^
2
2
x
x
2 2
1
x x
x
choose
arbitrary , 1
) (^2) (
2 2
) (^2) (
x
x^
c
c
x x
Normalized EigenvectorsNormalized
Eigenvectors
-^
From the previous example we see that eigenvectorsFrom the previous example, we see that eigenvectorsare determined up to a nonzero multiplicativeconstant.
-^
If this constant is specified in some particular way,then the eigenvector is said to be
normalized
-^
For example eigenvectors are sometimes normalized
-^
For example, eigenvectors are sometimes normalizedby choosing the constant so that ||
x || = (
x ,
x
Eigenvectors and Linear IndependenceEigenvectors
and Linear Independence
-^
If an eigenvalue
λ^
has algebraic multiplicity 1 then it is
If an eigenvalue
λ^
has algebraic multiplicity 1, then it is
said to be
simple
, and the geometric multiplicity is 1
also.
-^
If each eigenvalue of an
n
x
n
matrix
A
is
simple
, then
A
has
n
distinct eigenvalues. It can be shown that the
n^
eigenvectors corresponding to these eigenvalues
n^
eigenvectors corresponding to these eigenvalues
are linearly independent.
-^
If an eigenvalue has one or more
repeated
i^
l^
th
th
b^
f^
th
li^
l
eigenvalues
, then there may be fewer than
n
li
nearly
independent eigenvectors.
This may lead to
complications in solving systems of differential
p^
g^
y
equations.
Example 6: Eigenvalues
(1 of 5)
Example
6: Eigenvalues
(1 of 5)
-^
Find the eigenvalues and eigenvectors of the matrix
A
Find the eigenvalues and eigenvectors of the matrix
A
=^
A
-^
Solution: Choose
λ^
such that det(
A
-^ λ
I ) = 0, as follows.
⎜⎝^
(^
)^
det
det
λ I
A
2
3
⎜⎝^
3
2
1
2
