Q: How to Find eigen values and eigen vectors for a matrix?
A: This topic is very important for GATE, as one question is asked almost every year in which you are required to find out the eigen vector of a matrix.
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a) Eigen Value:
Suppose a matrix "P" is given. Now, eigen values can be found by simply solving the expression [P-λI]=0; for "λ", where I is identity matrix.
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For example: P= 5 3
1 3
Now, doing [P-λI]=0, we get, (5-λ)(3-λ)-3=0,
which gives λ=6,2
Now, these are eigen values of this matrix.
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b) Eigen Vector:
Now, in order to find out Eigen vector, select any one of the eigen values.
Now use the equation [P][x y]' = λ[x y]'
to find out the ratio of x and y.
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For example, we select λ=2.
Now, using the equation, we get 5x+3y=2x and x+3y=2y;
From each of the equations, we get x+y=0 or x=-y;
Hence, ratio of x and y, i.e. [1 -1] is our eigen vector..
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.Remember: Both the equations in second last step should give same eigen vector.
