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

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    Find the eigen values and eigen vectors of the matrix

    A = [9, -1 , 9; 3 , -1 , 3; -7 , 1 , -7]

    Also find the corresponding eigen spaces for the matrix A.
    Find a matrix B which reduce the given matrix A to the diagonal form by the transformation B^(-1)AB.

    The fully formatted problem is in the attached file.

    © BrainMass Inc. brainmass.com December 24, 2021, 5:24 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/eigen-values-eigen-vectors-matrix-45924

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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 .

    ------------------------------------------------------------------------------------------------------
    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.
    ---------------------------------------------------------------------------------------------

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 5:24 pm ad1c9bdddf>
    https://brainmass.com/math/linear-algebra/eigen-values-eigen-vectors-matrix-45924

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