Find the eigen values and eigen vectors of the matrix A = [9, -1, 9; 3, -1, 3; -7, 1, -7].

The solution the Posting is in the attached file.

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Linear Algebra
Matrices (VIII)
Eigen Values, Eigen Vectors, Eigen Spaces
and Diagonalizable Matrix

By:- Thokchom Sarojkumar Sinha

Find the eigen values and eigen vectors of the matrix

Also find the corresponding eigen spaces for the matrix .
Find a matrix which reduce the given matrix to the diagonal form by the transformation .

Solution :-

Then gives

or,

or,

or,

or,

or,

Therefore eigen values are and .

Consider the matrix equation

This gives --------------------------------(1)

Consider the case in which
Then (1) is reduced to
or,

i.e.,

This

On solving the first two equations, we get

or,

or, , say.

Then is an eigen vector corresponding to the eigen value ,
being any non-zero real number.

Next consider the case in which
Then (1) is reduced to

i.e.,

or,

This is equivalent to

i.e.,

Here the first and the last equations are identical.
On solving the first two equations, we get

or,

or, , say.

Then is an eigen vector corresponding to the eigen value ,
being any non-zero real number.

Again consider the case in which
Then (1) is reduced to

i.e.,

or,

This is equivalent to

On solving the first two equations, we get

or,

or, , say.

Then is an eigen vector corresponding to the eigen value ,
being any non-zero real number.

Hence the eigen vectors are

, and ;

Thus the subspaces spanned by the vectors

, and separately are the three eigen spaces of corresponding to the
eigen values .

Writing the eigen vectors as three columns of a matrix, we get a matrix

which reduce the given matrix to the diagonal form by the transformation .

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Note:- Diagonalization Matrix:
A matrix over a field is said to be diagonalizable if it is similar to a diagonal matrix over .
That is, the matrix is diagonalizable if there exists a non-singular matrix such that
diagonal matrix .
And the process is called to diagonalize or to transform to diagonal form.
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