Unitary and Orthogonal Operators

• Let be an inner product space over and let be a linear operator on . If

  For all .

  If we call a unitary operator. If we call an orthogonal operator.

  Similarly for matrix . If

  If , we call a unitary matrix If , we call an orthogonal matrix

• In other words Unitary operators preserve the lengths of vectors in the vector space.

• (Friedberg 6.18) Let be a finite dimensional inner product space and let be a linear operator on . Then the following are equivalent

  • (see Normal Matrix)
  • .
  • If is an orthonormal basis for , then is an orthonormal basis for .
  • There exists an orthonormal basis for such that is an orthonormal basis for .
  • for all .

• (Friedberg 6.18.1) Let be a linear operator on a finite-dimensional real inner product space . has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value if and only if is both self-adjoint and orthogonal.

  Another way to say this is that all eigenvalues of lie along the unit circle

• (Friedberg 6.18.2) Let be a linear operator on a finite dimensional complex inner product space. Then has an orthonormal basis of eigenvectors of with corresponding eigenvalues of absolute value if and only if is unitary.

  • If is orthogonal / unitary, then

• and are unitarily / orthogonally equivalent if and only if there exists a unitary / orthogonal matrix such that

  This is another form of similarity between matrices if we take the fact that

• (Friedberg 6.19) Let . Then is normal if and only if is unitarily equivalent to a diagonal matrix.

• (Friedberg 6.20) Let . Then is symmetric if and only if is orthogonally equivalent to a real diagonal matrix.

• (Friedberg 6.21) Schur’s Theorem (Matrix Form). Let and whose characteristic polynomial splits over

  • If then is a unitarily equivalent to a complex upper triangular matrix.
  • If then is an orthogonally equivalent to a real upper triangular matrix.

• (Friedberg e6.5.10) Let be a complex normal or real symmetric matrix with . We have that

  • Proof: is unitarily / orthogonally equivalent to a diagonal matrix, that is . The diagonal matrix has the eigenvalues as its entries. We have by virtue of having the same shape as
    Similarly,

• (Friedberg e6.5.12) Let be an real symmetric or complex normal matrix. Let . We have the determinant

  • Proof: is unitarily / orthogonally equivalent to a diagonal matrix, that is . The diagonal matrix has the eigenvalues as its entries. We have

Links §

• Friedberg, Insel and Spence