*eigenvectors*can be obtained. 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.

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Consider for example:

A =

(1.....i)

(-i.....1)

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We have P_A(t) =

(t -1.........i)

(-i.........t-1)

So: P_A(t) = (t - 1)

^{2}- (-i)(i) = t

^{2}-2t +1 -1 = 0

Then: P_A(t) = t

^{2}- 2t = t(t - 2)

The eigenvalues E1,2 are:

E1 = 0, E2 = 2

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:

A =

(1.. …. .2)

(2.......-2)

Then: P_A(t) =

(t - 1......2)

2........t +2)

P_A(t) = (t - 1)((t + 2) - 4

P_A(t) = t

^{2}+ t - 2 -4 = t

^{2}+ t - 6

But: t

^{2}+ t - 6 = (t + 3) (t - 2)

So: 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]

Working these linear algebra problems is fairly straightforward once one follows the steps such as I've outlined above.

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