Unformatted text preview: Notes on ERO operations and Eigenvalues and
Eigenvectors
Elementary Row Operations
Denition. An elementary row operation (ERO) on an augmented matrix pro duces a new augmented matrix corresponding to a new (but equivalent) system
of linear equations. Two matrices are said to be rowequivalent if one can be
obtained from the other by a nite sequence of elementary row operations. Theorem. There are
matrix. These are: 3 elementary row operations possible on an augmented 1. Interchanging two rows.
2. Multiplying a row by a nonzero constant.
3. Adding a multiple of a row to another row. Example. An example of each of the ERO's are given below.
1. Interchanging the 1st and 2nd rows. 0
−1
2 1
2
−3 3 4
0 3 =⇒
4 1 −1
0
2 2
1
−3 0 3
3 4
4 1 R1 ↔ R2 2. Multiply the 1st row by 2
1
5 −4
3
−2 1
2 6 −2
−3 0 =⇒
1
2
1
r1 → R1
2 to produce a new 1st row. 1
1
5 −2
3
−2 3
−3
1 −1
0
2 3. Add −2 times the 1st row to the 3rd row to produce a new 3rd row. 1 1 2
0 3
2 1 −4
−2
5 3
−1 =⇒
−2 1
0
0 2
3
−3 −4
−2
13 3
−1
−8 −2r1 + r3 → R3 Eigenvalues and Eigenvectors
Eigenvalues is one of the most important topics in Linear Algebra. Some applications of eigenvalues are studying populations growth, solving systems of
dierential equations, in quadratic forms, and in engineering and science. The
concept of eigenvalues comes from the eigenvalue problem. Eigenvalue Problem
If A is a n × n matrix, do there exist nonzero matrices x of size n × 1 such
that Ax is a scalar multiple of x? The fundamental equation to the eigenvalue
problem is
Ax = λx. where λ is the eigenvalue of A and x is the eigenvector of A.
1 4
1
2
Example. Let A = 2 3 , and x1 = 1 and x2 = −1
.
Verify that λ1 = 5 is an eigenvalue of A corresponding to x1 and λ2 = −1 is an
eigenvalue of A corresponding to x2 . Solution.
1 4 1
5
=
Ax1 =
2 3 1
5
1 4
2
−2
Ax2 =
=
2 3 −1
1
1
= λ 1 x1
1
2
= (−1)
= λ 2 x2 .
−1
= (5) Finding Eigenvalues and Eigenvectors
The procedure to nd the eigenvalues and its corresponding eigenvectors is given
below. Given the fundamental equations, we have
Ax = λx Ax = λIx O = λIx − Ax O = (λI − A)x which is a homogenous system. The homogenous system has nonzero solutions
if and only if the coecient matrix (λI − A) is singular; that is, if and only
if, the determinant of λI − A = 0. The equation λI − A = 0 is called the
characteristic equation of A.
2 Example. Find the eigenvalues and eigenvectors of the matrix A =
1
2
4
.
3 Solution. The characteristic equation of A is
1 0
1 4 0 = λI − A = λ
−
0 1
2 3
λ 0
1 4 −
0 = 0 λ
2 3 λ − 1
−4 0 = −2
λ − 3 (λ − 1)(λ − 3) − (−2)(−4) 0 = 0 = λ2 − 4λ − 5 0 = (λ − 5)(λ + 1). This yields two eigenvalues, λ1 = 5 and λ2 = −1.
We now go on to nd the eigenvectors of each of the eigenvectors.
For λ1 = 5, substitute into the homogenous system, we have
λ−1
−2 −4
λ−3
0
0 =⇒
=
(5) − 1
−4
0
−2
(5) − 3 0
4 −4 0
.
−2 2 0 Then rowreducing the system, we have
4 −4 0
−2 2 0
1 −1 0
−2 2 0
1 −1 0
.
0 0 0 1
r1 → R1
4
2r1 + r2 → R2 The row of zeros tells us that the system has innitely many solutions
and thus has to be represented by parametric representation. The
system in equation form is x1 − x2 = 0.
Let x2 = t, t ∈ R. Then x1 = x2 = t as well. Then the eigenvectors x is
given by
x
x= 1
x2
t
1
=
=t
, t ∈ R.
t
1 Thus, all eigenvectors corresponding to λ1 = 5 are nonzero scalar multiples of
1
.
1 3 For λ1 = −1, substitute into the homogenous system, we have
λ−1
−2 −4
λ−3
0
0
(−1) − 1
−4
0
−2
(−1) − 3 0
−2 −4 0
.
−2 −4 0 =⇒
= Then rowreducing the system, we have
−2 −4 0
−2 −4 0
1
2 0
−2 −4 0
1 2 0
.
0 0 0 1
− r1 → R1
2
2r1 + r2 → R2 The system in equation form is x1 + 2x2 = 0.
Let x2 = t, t ∈ R. Then x1 = −2x2 = −2t as well. Then the eigenvectors x
is given by
x=
x1
=
x2
−2t
−2
=t
, t ∈ R.
t
1 Thus, all eigenvectors corresponding to λ1 = −1 are nonzero scalar multiples
of
2
.
1 Exercise. Find the eigenvalues and corresponding eigenvectors of the matrix 1
A= 1
−1 2
2
−1 −2
1 .
0 Solution. The eigenvalues and corresponding eigenvectors are −2
x1 = 1 1 2
x2 = −1
1 −2
x3 = −1 .
1 λ1 = 1 λ2 = −1 λ3 = 3 4 ...
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 Winter '16
 Alveen Chand
 Linear Algebra, Eigenvectors, elementary row operations

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