Some Special Matrices

    A. Define symmetric matrix, skew-symmetric matrix, and orthogonal
        matrix. Prove identities involving these types of matrices.

    B. Characterize and determine the eigenvalues and eigenvectors of
        symmetric, skew-symmetric, and orthogonal matrices. Derive basic facts concerning these
        matrices.

    C. Define an orthonormal set of vectors. Determine
        whether a set of vectors is orthonormal.

    D. Relate the orthogonality of a matrix to the orthonormality of its
        column (or row) vectors.

    E. Diagonalize a symmetric matrix.

Lecture:

    1. Define for matrices
        a. symmetric
        b. skew-symmetric
        c. orthogonal
    2. Discuss the special properties of their eigenvalues.
        a. symmetric - all eigenvalues are real.
        b. skew-symmetric - eigenvalues are pure imaginary or zero.
        c. orthogonal - eigenvalues are real or complex conjugate
            pairs and have absolute value equal to 1.
    3. Prove each of the above properties.
    4. Prove that every real matrix can be written as the sum of a
        symmetric matrix and a skew-symmetric matrix.
    5. Interpret an orthogonal matrix as a transformation that is a
        rotation or a rotation combined with a reflection.
    6. Conjecture how a.b compares to Aa . Ab if A is
        an orthogonal matrix. Prove this conjecture.
    7. Conjecture how a compares to Aa if A is
        an orthogonal matrix. Prove this conjecture.
    8. Prove that the determinant of an orthogonal matrix is 1 or -1.
    9. Prove that the column or row vectors of an orthogonal
        matrix form an orthonormal set.