Matrix operations, complex

Most of the discussion in this book concentrates on matrices whose elements are real numbers, these being relevant for most applications. However, most of what is described works equally well for complex and real elements, if one observes the following formal changes:

• the transpose File:Hepa img659.gif becomes the conjugate (Hermitian) transpose File:Hepa img660.gif ;
• the inner product File:Hepa img661.gif becomes File:Hepa img662.gif ;
• orthogonality File:Hepa img663.gif is written x^H y = 0;
• the length File:Hepa img664.gif becomes File:Hepa img665.gif ;
• a symmetric matrix becomes Hermitian A^H = A;
• an orthogonal matrix or File:Hepa img666.gif becomes unitary U^HU=I or U^H = U^-1; hence File:Hepa img667.gif becomes (Ux)^H(Uy) = x^Hy, and File:Hepa img668.gif remains File:Hepa img669.gif .

For further reading, see Strang88.