- #1

Number2Pencil

- 208

- 1

## Homework Statement

Find the eigen values/eigen vectors of

[tex]

\left[

\begin{array}{cc}

3&-1&1&1&0&0\\

1&1&-1&-1&0&0\\

0&0&2&0&1&1\\

0&0&0&2&-1&-1\\

0&0&0&0&1&1\\

0&0&0&0&1&1\\

\end{array}

\right]

[/tex]

## Homework Equations

## The Attempt at a Solution

The lambda equation is:

[tex]

A - \lambda I =

\left[

\begin{array}{cc}

3 - \lambda&-1&1&1&0&0\\

1&1 - \lambda&-1&-1&0&0\\

0&0&2 - \lambda&0&1&1\\

0&0&0&2 - \lambda&-1&-1\\

0&0&0&0&1 - \lambda&1\\

0&0&0&0&1&1 - \lambda\\

\end{array}

\right]

[/tex]

Finding the determinant yields:

[tex]

-32 \lambda + 80\lambda^2 - 80\lambda^3 + 40\lambda^4 - 10\lambda^5 + \lambda^6

[/tex]

Setting equal to zero and factoring gives the eigen values:

[tex]

\lambda =

\left[

\begin{array}{cc}

0\\

2\\

2\\

2\\

2\\

2\\

\end{array}

\right]

[/tex]

Plugging in lambda = 2 into the lambda equation gives:

[tex]

\left[

\begin{array}{cc}

1&-1&1&1&0&0\\

1&-1&-1&-1&0&0\\

0&0&0&0&1&1\\

0&0&0&0&-1&-1\\

0&0&0&0&-1&1\\

0&0&0&0&1&-1\\

\end{array}

\right]

[/tex]

The columns help for the simultaneous equations:

The bottom four rows reveal that X5, X6 = 0

Adding the first two rows together gives:

2X1 - 2X2 = 0

or

X1 = X2, So I choose X1 = X2 = 1, since the eigen vector must be a non-zero vector.

X3 and X4 cancel out so I choose them to = 0.

-------------

Since the eigen values repeat, I have to use the previous eigen vector as the answer to the simultaneous equations:

Summing the top two rows together:

2X1 - 2X2 = 2

or

X1 = 1 + X2. Everything else is still zero, choose X2 = 0, X1 = 1

--------------

Recursively using the previous eigen vector:

2 equations, 4 unknowns, choose X2 = 0, X4 = 0, top two equations become:

X1 + X3 = 1

X1 - X3 = 0

X1 and X3 = 1/2

--------------

Recursively using the previous eigen vector:

This is where the trouble comes into play:

3rd equation : X5 + X6 = 1/2

4th equation: -X5 - X6 = 0

which cannot be solved...

This keeps happening over and over. I used MATLAB to verify that the eigen values and vectors could be found...but I cannot solve this through.

Any suggestions?