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    To begin with, here is a simple lemma.

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    The eigenvalues of a real symmetric matrix are real.

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    PsX p s as there are columns to the left of I PsX p s in those two inside matrices.

    The next major result is called the Fredholm alternative.

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    Sometimes sequences come in terms of a recursion formula.

    Here are some matrices.

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    Let B be a q x p block matrix as in 13.

    Gram Schmidt theorem to the resulting basis.

    Then what rank 1 matrix would be closest to this one in the Frobenius norm?

    If the matrix is orthogonal, determine whether it is proper or improper.

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    Proof: For x the minimizer of Theorem 13.

    Let A be a real symmetric matrix.

    Thus A is pure imaginary.

    In the following example, this orthogonal matrix will be found.

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    This is really obvious if you look at a simple example.

    Is U an orthogonal matrix?

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    Show this can be considered as a discreet dynamical system as follows.

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    Orthogonal matrices are divided into two classes, proper and improper.

    Proof: The proof is given later.

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    Note that every real orthogonal matrix is unitary.

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    Let A be a Hermitian matrix such that all its eigenvalues are nonnegative.

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    The best bases are orthonormal.

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    However, you will show in Problem 23 that an orthonormal set of n vectors in R n is always a basis.

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    Suppose U is an orthogonal matrix.

    Here is another lemma about preserving distance.

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    This is easy to see from the properties of determinants.

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    It remains to find the third vector in the orthonormal basis.

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    As a simple consequence of the above theorem, here is an interesting lemma.

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    Therefore, since there are three of these vectors, they must constitute a basis.

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    A' is the one which is closest to A in the Frobenius norm.

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    With this preparation, here is the big theorem about the right polar factorization.

    AU is a diagonal matrix.

    Proof: This follows from Theorem 13.

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    This is left for you to do.

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    The following corollary is also called the Fredholm alternative.

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    Proof: By the above lemma and Theorem 13.

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    Not all bases for F n are created equal.

    However, this is as far as I will go with this in this part of the book.

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    The process by which these vectors were generated is called the Gram Schmidt process.

    Why is the singular value decomposition important?

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    This is what it means for the set of vectors to be linearly independent.

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    With this definition and lemma here is the main theorem on the singular value decomposition.

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    It remains to find an orthonormal basis.

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    An important application of Corollary 13.

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    Remember that to find the inverse of a matrix was often a long process.

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    It remains to verify this works.

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    Proof: Since A is symmetric, then by Theorem 13.

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    Therefore, from Lemma 13.

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    Some matrices are non defective and some are not.

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    Then by Theorem 13.

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    Recall the definition of the adjoint of a matrix.

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    Proof: This follows from the definition.

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    Therefore, by Lemma 13.

    This is because the columns (rows) form an orthonormal set of vectors.

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    This proves half the theorem.

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    This is a symmetric matrix.

    Note that this all makes sense.

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    Here is an interesting and significant application of block multiplication.

    This turns out to be very useful.

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    As indicated in Theorem 13.

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    Consider the following product.

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    This means you take the transpose of A and then replace each entry by its conjugate.

    Next consider the question of multiplication of two block matrices.

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    Proof: Suppose first A is onto.

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    This lemma immediately implies the following lemma.

    Suppose now that z is some other solution to this least squares problem.

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    Thus the block Aij in this case is a matrix of size Si x pj.

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    Then from Lemma 13.

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    Also note that by Lemma 13.

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    In this section, A will be an m x n matrix.

    The Frobenius norm is one of many norms for a matrix.

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    This is really easy to do using a computer algebra system.

    This is a skew symmetric matrix.

    Consider the subspace U (F n ).

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    Every matrix is related to an upper triangular matrix in a particularly significant way.

    Therefore, continuing this way, the process of the proof delivers only real vectors and real matrices.

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    However, here is a proof in this special case.

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