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    Matrices Eigenvalues, eigenspace, and eigenbasis.

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    Matrices Eigenvalues, eigenspace, and eigenbasis.

    See attached file.

    For each of the matrices, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis, if you can.
    [■(7&[email protected]&9)]
    [■(1&[email protected]&1)]
    [■(6&[email protected]&7)]
    [■(0&[email protected]&2)]
    [■(1&1&[email protected]&2&[email protected]&0&3)]
    [■(1&1&[email protected]&1&[email protected]&0&1)]
    [■(■(0&[email protected]&1)&■(0&[email protected]&1)@■(0&[email protected]&0)&■(0&[email protected]&1))]

    © BrainMass Inc. brainmass.com December 24, 2021, 10:08 pm ad1c9bdddf
    https://brainmass.com/math/linear-algebra/matrices-eigenvalues-eigenspace-eigenbasis-446911

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    SOLUTION This solution is FREE courtesy of BrainMass!

    1.
    The matrix is:
    (1.1)
    This is an upper triangular matrix, hence the eigenvalues are the elements of the diagonal:
    (1.2)
    The first eigenvector:

    (1.3)
    The second eigenvector:

    (1.4)
    We have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis.

    2.
    The matrix is:
    (2.1)
    The eigenvalues are:

    (2.2)
    The first eigenvector:

    Therefore:
    (2.3)
    The second eigenvector:

    (2.4)
    As before we have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis.

    3.
    The matrix is:
    (3.1)
    The eigenvalues are:

    (3.2)
    The first eigenvector:

    (3.3)

    The second eigenvector:

    (3.4)

    As before we have two distinct eigenvalues and two distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these two eigenvectors form an eigenbasis.

    4.

    The matrix is:
    (4.1)
    The eigenvalues:

    (4.2)
    For the eigenvectors we have a single eigenvalue, hence:

    We have a single eigenvector
    (4.3)
    We have a single eigenvalue with algebraic multiplicity , but with a single associated eigenvector (geometric multiplicity ) therefore this single vector cannot be an eigenbasis.
    5.
    The matrix is:
    (5.1)
    This is an upper triangular matrix, hence the diagonal elements are the eigenvalues:
    (5.2)
    The first eigenvector:

    (5.3)

    The second eigenvector:

    (5.4)
    the third eigenvector:

    (5.5)
    We have three distinct eigenvalues and three distinct eigenvectors. The algebraic multiplicity (m=1) of each of the eigenvalues equal their respective geometric multiplicity, hence these three eigenvectors form an eigenbasis.

    6.
    The matrix is:
    (6.1)
    This is an upper triangular matrix, hence the diagonal elements are the eigenvalues:
    (6.2)
    Since the matrix is in a canonical Jordan form we know it has no eigenbasis.
    The single eigenvector is:

    (6.3)
    Obviously, we cannot span with a single vector. therefore, this eigenvector does not form an eigenbasis.

    7.
    The matrix is:
    (7.1)
    It is an upper triangular matrix hence the eigenvalues are the diagonal elements are the eigenvalues:
    (7.2)
    This is an obviously a defective matrix, hence its eigenvectors do not form an eigenbasis.
    The first two eigenvectors:

    (7.3)

    The third eigenvector:

    (7.4)
    We have two eigenvalues (each with algebraic multiplicity ) and for the geometric multiplicity of the eigenvalue is 2 as well.
    However the geometric multiplicity of is only 1, hence the matrix is defective and has no eigenbasis.

    a.
    The matrix is:
    (8.1)
    This is a lower triangular matrix hence the eigenvalues are the diagonal elements:
    (8.2)

    b.
    The matrix is:

    This is a block-diagonal matrix. Its diagonal is composed of two matrices, hence its determinant is the product of the matrices' determinants.

    The eigenvalues are:

    (8.3)

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

    © BrainMass Inc. brainmass.com December 24, 2021, 10:08 pm ad1c9bdddf>
    https://brainmass.com/math/linear-algebra/matrices-eigenvalues-eigenspace-eigenbasis-446911

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