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?

Question

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?

asijikisi67 · Accepted Answer

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&#039;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.

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?

Answered question

Answer & Explanation

New Questions in Linear algebra