One of the key reasons for which my class was taught Linear Algebra, including how to compute Eigenvalues and Eigenvectors of Matrices, was so that we could Diagonalize Symmetrical Matrices, in Real Numbers. What this did was to compute the ‘Perpendicular Matrix’ of a given matrix, in which each column was one of its Eigenvectors, and which was an example of an Orthogonal Matrix. (It might be the case that what was once referred to as a Perpendicular Matrix, may now be referred to as the Orthogonal Basis of the given matrix,?)
Having computed the perpendicular matrix P of M, it was known that the matrix product
PT M P = D,
which gives a Diagonal Matrix ‘D’. But, a key problem my Elementary Linear class was not taught to solve, was what to do if ‘M’ had complex Eigenvalues. In order to be taught that, we would need to have been taught in general, how to combine Linear Algebra with Complex Numbers. After that, the Eigenvectors could have been computed as easily as before, using Gauss-Jordan Elimination.
I have brushed up on this in my old Linear Algebra textbook, where the last chapter writes about Complex Numbers. Key facts which need to be understood about Complex Vector Spaces, is
- The Inner Product needs to be computed differently from before, in a way that borrows from the fact that complex numbers naturally have conjugates. It is now the sum, of each element of one vector, multiplied by the conjugate, of the corresponding element of the other vector.
- Orthogonal and Symmetrical Matrices are relatively unimportant with Complex Elements.
- A special operation is defined for matrices, called the Conjugate Transpose, A* .
- A Unitary Matrix now replaces the Orthogonal Matrix, such that A-1 = A* .
- A Hermitian Matrix now replaces the Symmetrical Matrix, such that A = A* , and the elements along the main diagonal are Real. Hermitian Matrices are also easy to recognize by inspection.
- Not only Hermitian Matrices can be diagonalized. They have a superset, known as Normal Matrices, such that A A* = A* A . Normal Matrices can be diagonalized.
This could all become important in Quantum Mechanics, considering the general issue known to exist, by which the bases that define how particles can interact, somehow need to be multiplied by complex numbers, to describe accurately, how particles do interact.