# Find the eigenspace of this 2x2 matrix????????????????????????

[5 -2 ]

[8 13]

has one eigenvalue of multiplicity 2. Find this eigenvalue and the dimenstion of the eigenspace.

eigenvalue

dimension of the eigenspace = ??????????????????????????????????????????????

Relevance

Let A be the given matrix

[5 -2 ]

[8 13].

To find the eigenvalue(s) lambda, solve det(A - lambda I) = 0.

Note that A - lambda I is the matrix

[5-lambda -2]

[8 13-lambda].

0 = det(A - lambda I) = (5-lambda) (13-lambda) + 16

= 81 - 18lambda + lambda^2

= (lambda - 9)^2.

So the eigenvalue is lambda = 9, with multiplicity 2.

For an eigenvalue lambda, the eigenspace is the solution set of the equation (A - lambda I)x = 0.

For lambda = 9, (A - lambda I) becomes the matrix

[-4 -2]

[8 4].

So (A - lambda I)x = 0 becomes the system

-4x1 - 2x2 = 0

8x1 + 4x2 = 0.

These two equations are equivalent (redundant), since the second equation is just -2 times the first equation. So we can consider just the first equation, which gives

x1 = x1

x2 = -2x1

and so there is one free variable, x1. Thus the dimension of the eigenspace (solution set) is 1.

(By the way, this tells us that the original matrix, A, is not diagonalizable, since there is at least one eigenvalue for which the dimension of the eigenspace is less than the multiplicity.)

Lord bless you today!

• The characteristic polynomial equals (5-L)(13-L)+16 = L^2-18L+81 = (L-9)^2. Therefore the only eigenvalue is 9. To find the corresponding eigenspace we have to solve the following system:

-4x-2y = 0

8x+4y = 0.

Since the equations are proportional the eigenspace is the line 2x+y=0 and has therefore dimension 1.