Number2Pencil
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Homework Statement
Find the eigen values/eigen vectors of
<br /> \left[<br /> \begin{array}{cc}<br /> 3&-1&1&1&0&0\\<br /> 1&1&-1&-1&0&0\\<br /> 0&0&2&0&1&1\\<br /> 0&0&0&2&-1&-1\\<br /> 0&0&0&0&1&1\\<br /> 0&0&0&0&1&1\\<br /> \end{array}<br /> \right]<br />
Homework Equations
The Attempt at a Solution
The lambda equation is:
<br /> A - \lambda I =<br /> \left[<br /> \begin{array}{cc}<br /> 3 - \lambda&-1&1&1&0&0\\<br /> 1&1 - \lambda&-1&-1&0&0\\<br /> 0&0&2 - \lambda&0&1&1\\<br /> 0&0&0&2 - \lambda&-1&-1\\<br /> 0&0&0&0&1 - \lambda&1\\<br /> 0&0&0&0&1&1 - \lambda\\<br /> \end{array}<br /> \right]<br /> <br />
Finding the determinant yields:
<br /> -32 \lambda + 80\lambda^2 - 80\lambda^3 + 40\lambda^4 - 10\lambda^5 + \lambda^6<br />
Setting equal to zero and factoring gives the eigen values:
<br /> \lambda = <br /> \left[<br /> \begin{array}{cc}<br /> 0\\<br /> 2\\<br /> 2\\<br /> 2\\<br /> 2\\<br /> 2\\<br /> \end{array}<br /> \right]<br />
Plugging in lambda = 2 into the lambda equation gives:
<br /> \left[<br /> \begin{array}{cc}<br /> 1&-1&1&1&0&0\\<br /> 1&-1&-1&-1&0&0\\<br /> 0&0&0&0&1&1\\<br /> 0&0&0&0&-1&-1\\<br /> 0&0&0&0&-1&1\\<br /> 0&0&0&0&1&-1\\<br /> \end{array}<br /> \right]<br />
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.
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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
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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
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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?