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 .

Rayleigh Conditioning

Let be a self-adjoint matrix. The Rayleigh quotient for is defined as the scalar
(Friedberg 6.36) For a self adjoint matrix , is the largest eigenvalue of . Similarly, is the smallest eigenvalue of .
- Proof. Choose an orthonormal basis consisting of eigenvectors of . Represent
  
  Assume that the eigenvalues are sorted .
  
  Computing , we have
  
  We can clearly bound this as

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