If the matrix of a linear transformation: R^N->R^N with respect to some basis is symmetric, what does it say about the transformation?

Chelsea Lamb 2022-09-26 Answered
If the matrix of a linear transformation : R N R N with respect to some basis is symmetric, what does it say about the transformation? Is there a way to geometrically interpret the transformation in a nice/simple way?
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Answers (1)

asijikisi67
Answered 2022-09-27 Author has 10 answers
If R n is endowed with an inner product , and the matrix A of T is symmetric with respect to an orthonormal basis, then we have the important property that
y , T ( x ) = y , A x = y A x = y T A T x = ( A y ) T x = A y , x = T ( y ) , x ;
in this case we say that T itself is symmetric. There's too much to say about why these are important in a single post, but let me point out two useful facts:
(1) By the Spectral Theorem, T is orthogonally diagonalizable, that is, T is conjugate by an orthogonal transformation to a diagonal transformation.
(2) Suppose x , y are eigenvectors of T. If they correspond respectively to distinct eigenvalues λ , μ, then we have
λ x , y = λ x , y = T ( x ) , y = x , T ( y ) = x , μ y = μ x , y .
In particular, if λ μ then x , y = 0, that is, the eigenspaces of T are all orthogonal.
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