Self-Adjoint Matrix

• Let be an inner product space and let be a linear operator on . is self-adjoint if

  Similarly, a matrix is self-adjoint if

  We also call these matrices Hermitian Matrices

• An important example of such a matrix is a symmetric matrix where .

• (Friedberg Lem.6.17.1) Let be a self-adjoint operator on a finite-dimensional inner product space . Then

  • Every eigenvalue of is real. That is
  • The characteristic polynomial of splits.

• (Friedberg 6.17) Let be a linear operator on a finite dimensional real inner product space . Then is self-adjoint if and only if there exists an orthonormal basis of eigenvectors of .

• A similar class of matrices is the skew-adjoint matrices. Defined such that

  • An important example of such a matrix is a skew-symmetric matrix where
  • Let be a skew-adjoint operator on a finite-dimensional inner product space on . Then
    • Every eigenvalue is pure imaginary. That is
    • The characteristic polynomial splits.

• Let . is a Gramian matrix if there exists a real square matrix such that

• (Friedberg e6.4.12) is a Gramian matrix If and only if is symmetric and all its eigenvalues are nonnegative.

• (Friedberg Lem.6.18) Let be a finite dimensional inner product space and let be a self adjoint operator on . If then .

Link §

• Linear Algebra by Friedberg Insel and Spence