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Math Help - Symmetric Matrices

  1. #1
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    Symmetric Matrices

    Show that every symmetric matrix A=A^T can be written as U=UDU^T where D is a diagonal matrix and UU^T=I (orthogonal:unitary)

    i am pretty sure that this should be tried through induction.
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  2. #2
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    Quote Originally Posted by steiner View Post
    Show that every symmetric matrix A=A^T can be written as U=UDU^T (A=UDU^T ?) where D is a diagonal matrix and UU^T=I (orthogonal:unitary)
    i am pretty sure that this should be tried through induction.
    Lemma 1. The eigenvalues of a symmetric matrix with real entries are real numbers.

    Proof. If \lambda is an eigenvalue and v a corresponding eigenvector of an n \times n symmetric matrix A.
    Then Av = \lambda v . If we multiply each side of this equation on the left by v^T, then we obtain
    v^TAv = v^T(\lambda v) = \lambda v^T v = \lambda ||v||^2 . Then,

    \lambda = \frac{v^TAv}{||v||^2} .

    Thus, \lambda is a real number. To further verify, you apply a conjugate transpose and

    ({v^TAv})^* = v^TA^T(v^T)^T = v^TAv , since A is a symmetric matrix.

    Lemma 2. Let A be a real symmetric n \times n matrix. There is an orthogonal matrix U \in O_n(\mathbb{Re}) such that U^TAU is diagonal.

    Proof. We first show that eigenvectors from different eigenspaces with respect to a symmetric matrix A are orthogonal.
    Let v_1 and v_2 be eigenvectors corresponding to distinct real eigenvalues \lambda_1 and \lambda_2 of the matrix A. We shall show that v_1 \cdot v_2 = 0. Since A is a symmetric matrix, Av_1 \cdot v_2 = v_1 \cdot A^Tv_2 = v_1 \cdot Av_2 . This implies that \lambda_1 v_1 \cdot v_2 = v_1 \cdot \lambda_2 v_2 \Longleftrightarrow (\lambda_1 - \lambda_2)(v_1 \cdot v_2) = 0. Since \lambda_1 and \lambda_2 are distinct by hypothesis, v_1 \cdot v_2 = 0 .

    The above showed that eigenvectors from different eigenspaces are orthogonal. By applying a Gram-Schmit process, eigenvectors obtained within same eigenspaces become orthonormal. This ensures that there exists an orthogonal matrix P \in O_n(R) whose columns are normalized eigenvectors of a symmetric matrix A.

    Let U= <br />
\begin{bmatrix}v_{11} & v_{12} & \cdots & v_{1n} \\ v_{21} &v_{22} & \cdots & v_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ v_{n1} & v_{n2}&v_{n3} & v_{n4}\end{bmatrix}
    be an orthogonal matrix whose columns are eigenvectors v_k= \begin{bmatrix}v_{1k} \\ v_{2k}\\ \vdots <br />
\\v_{nk} \end{bmatrix} of A.

    Since Av_1 = \lambda_1 v_1, Av_2 = \lambda_2 v_2, ... ,Av_n = \lambda_n v_n,

    <br />
AU=\begin{bmatrix}\lambda_1 v_{11} & \lambda_2 v_{12} & \cdots & \lambda_n v_{1n} \\ \lambda_1 v_{21} & \lambda_2 v_{22} & \cdots & \lambda_n v_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ \lambda_1 v_{n1} & \lambda_2 v_{n2} & v_{n3} & \lambda_n v_{n4}\end{bmatrix} = \begin{bmatrix}v_{11} & v_{12} & \cdots & v_{1n} \\ v_{21} &v_{22} & \cdots & v_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ v_{n1} & v_{n2}&v_{n3} & v_{n4}\end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & \cdots & 0 \\ 0 &\lambda_2 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots \\ 0 & 0& \cdots & \lambda_n \end{bmatrix}=UD.

    Thus, D=U^TAU and A=UDU^T, since U is an orthogonal matrix

    To apply an induction, assume n \times n symmetric matrix A holds the above lemma and show (n+1) \times (n+1) symmetric matrix holds the above lemma as well. The construction shown in the lemma 2 can be used without much modification.
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