Maths - Eigenvectors and Eigenvalues of 3×3 Matrix

The method for symbolic computation of eigenvectors and eigenvalues involves first finding the eigenvalues from the characteristic polynomial:

det(M - λ I) = 0

where, in this case for three dimensional matrix M =

m00 m01 m02
m10 m11 m12
m20 m21 m22

which gives the characteristic equation:

-λ³ + λ²(m00 + m11 + i) + λ(m01 m10 + m02 m20 + m21 m12 - m00 m11 - i m00 - i m11) + (m00 m11 i - m00 m12 m21 - m01 m10 i + m10 m02 m21 + m20 m01 m12 - m20 m02 m11) = 0

We then need to solve this cubic equation to give uto 3 values of λ.

We can then substitute these into

m00 - λ m01 m02
m10 m11 - λ m12
m20 m21 m22 - λ
=
0
0
0

which is equivalent to solving 3 simultaneous equations.

Example - 3D Rotation Matrix

As descrtibed on this page. A 3D rotation matrix can be written:

[R] = c*
1 0 0
0 1 0
0 0 1
+ t*
x*x x*y x*z
x*y y*y y*z
x*z y*z z*z
+s*
0 -z y
z 0 -x
-y x 0

where,

The eigenvalues are: e±iθ and 1 and the eigenvector is [x,y,z].

To check this, try λ=1:

c*
1 0 0
0 1 0
0 0 1
+ t*
x*x x*y x*z
x*y y*y y*z
x*z y*z z*z
+s*
0 -z y
z 0 -x
-y x 0
1 0 0
0 1 0
0 0 1
x
y
z
=
0
0
0

giving:

c*
1 0 0
0 1 0
0 0 1
+ t*
x*x x*y x*z
x*y y*y y*z
x*z y*z z*z
+s*
0 -z y
z 0 -x
-y x 0
-
1 0 0
0 1 0
0 0 1
x
y
z
=
0
0
0

now substitute t =1 - c:

(c-1)*
1 0 0
0 1 0
0 0 1
+ (1-c)*
x*x x*y x*z
x*y y*y y*z
x*z y*z z*z
+s*
0 -z y
z 0 -x
-y x 0
x
y
z
=
0
0
0

giving:

(1-c)*
-y²-z² x*y x*z
x*y -x²-z² y*z
x*z y*z -x²-y²
+s*
0 -z y
z 0 -x
-y x 0
x
y
z
=
0
0
0

multiplying out the matrix by the vector gives:

(1-c)*
(-y²-z²)*x+x*y²+x*z²
x²*y+(-x²-z²)y+y*z²
x²*z+y²*z+(-x²-y²)*z
+s*
-z*y+y*z
z*x-x*z
-y*x+x*y
=
0
0
0

Now try λ=e= c + i s

t*x*x + c - λ t*x*y - z*s t*x*z + y*s
t*x*y + z*s t*y*y + c - λ t*y*z - x*s
t*x*z - y*s t*y*z + x*s t*z*z + c - λ
x
y
z
=
0
0
0

Program

There are a number of open source programs that can calculate eigenvalues and eigenvectors. I have used Axiom, how to install Axiom here.

To get a numeric solution for a given matrix, we can use eigenvalues(m) and eigenvectors(m) as shown here:

I have put user input in red:

(1) -> m := matrix[[1,4,7],[2,5,8],[3,6,9]]
         +1  4  7+
         |       |
    (1)  |2  5  8|
         |       |
         +3  6  9+
 Type: Matrix Integer
(2) -> ev := eigenvalues(m)
               2
(2)  [0,%A | %A  - 15%A - 18]


Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol,
Polynomial Integer))
(3) -> eigenvectors(m)
 (3)
                                + 1 +
                                |   |
[[eigval= 0,eigmult= 1,eigvec= [|- 2|]],
                                |   |
                                + 1 +
                                                     +%G - 12+
                                                     |-------|
                                                     |   6   |
                  2                                  |       |
 [eigval= (%G | %G  - 15%G - 18),eigmult= 1,eigvec= [|%G - 6 |]]]
                                                     |------ |
                                                     |  12   |
                                                     |       |
                                                     +   1   +
Type: List Record(eigval: Union(Fraction Polynomial
Integer,SuchThat(Symbol,Polynomial Integer)),eigmult:
NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer)
       

Or we can find a general formula for a given matrix as shown here:

(1) -> msymb := matrix[[a,b,c],[d,e,f],[g,h,i]] 
        +a  b  c+
        |       |
(1)     |d  e  f|
        |       |
        +g  h  i+
Type: Matrix Polynomial Integer
       
(2) -> evsymb := eigenvalues(msymb)
 (2)
[
         %B
         |
                            2
((a - %B)e - b d - %B a + %B )i + ((- a + %B)f + c d)h
         +
                                  2                2      3
(b f - c e + %B c)g + (- %B a + %B )e + %B b d + %B a - %B
         ]
Type: List Union(Fraction Polynomial Integer,
   SuchThat(Symbol,Polynomial Integer))
(3) -> eigenvectors(msymb)
 (3)
         [
         [
eigval =
         %H
         |
                                     2
         ((a - %H)e - b d - %H a + %H )i + ((- a + %H)f + c d)h
         +
                                           2                2      3
         (b f - c e + %H c)g + (- %H a + %H )e + %H b d + %H a - %H
         ,
eigmult= 1,
         +                         2                     2             +
         | ((e - %H)h + b g)i - f h  + (- c g - %H e + %H )h - %H b g  |
         | ----------------------------------------------------------  |
         |                    2                     2                  |
         |                 d h  + (- e + a)g h - b g                   |
         |                                                             |
eigvec= [|                                            2             2  |]]
         |(- d h + (- a + %H)g)i + (f g + %H d)h + c g  + (%H a - %H )g|
         |-------------------------------------------------------------|
         |                     2                     2                 |
         |                  d h  + (- e + a)g h - b g                  |
         |                                                             |
         +                              1                              +
         ]
Type: List Record(eigval: Union(Fraction Polynomial Integer,
SuchThat(Symbol,Polynomial Integer))
,eigmult: NonNegativeInteger,eigvec:
List Matrix Fraction Polynomial Integer)
    

 


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