In the previous instalments on linear algebra, we saw how one can obtain the characteristic polynomial from a matrix as well as the eigenvalues. We now want to use past examples to show how the eigenvectors can be obtained. Consider again the matrix given in the first problem:
A =
(1.....i)
(-i.....1)
for which we found the eigenvalues were:
E1 = 0, E2 = 2, viz.
http://brane-space.blogspot.com/2011/12/solutions-to-linear-algebra-problems.html
To get the eigenvectors is just straightforward and merely requires obtaining simultaneous equations in x, y for example - based on using the rows in the matrix and applying each of the eigenvalues to them.
For example, take E1 = 0, then the resulting equations are:
x - iy = 0
ix + y = 0
or x = -iy and y = ix
The eigenvector is easily solved for and is:
v1 =
[-i]
[1]
Next, take the eigenvalue E2 = 2, then the simultaneous equations from the matrix A are:
x + iy = 2x
-ix + y = 2y
which yields: x = iy and y = -ix
Or, an eigenvector of: v2 =
[1]
[-i]
Consider now problem (2) which featured,
A =
(1.. …. .2)
(2.......-2)
And for which we found: E1 = -3, and E2 = 2
To get the eigenvector associated with the eigenvalue E1 = -3, we form the left side of the algebraic equations in x, and y using A such that:
x + 2 y = -3x
2x - 2y = -3y
Simplifying:
4x + 2y = 0
2x + y = 0
or:
4v1 + 2v2 = 0
2v1 + v2 = 0
and solving the simultaneous eqn. yields: v2 = - 2v1, or
v =
[1]
[-2]
For E2 = 2, we may write:
x + 2y = 2x
2x -2y = 2y
Again, the eigenvalue (E2) is always multiplied by x and then y to give the column matrix comprising the right side, with x-value on top, and y-value on the bottom. Simplifying the preceding equations:
-x + 2y = 0
2x - 4y = 0
Or:
-v1 + 2v2 = 0
2v1 - 4v2 = 0
which yields: v1 = 2v2, so the eigenvector in this case is:
v =
[2]
[1]
Problems:
1) Find the eigenvectors associated with the matrix A in Problem (3), e.g. for
A =
(3.. ……2)
(-2...... 3)
with the eigenvalues: E1 = 3 + 2i, and E2 = 3 - 2i
(2) Given the matrix:
A =
(2.......4)
(5.......3)
Find: a) the characteristic polynomial, b) the eigenvalues associated with it, and c) the eigenvectors.
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