Chapter I: Introduction to Matrix Calculations

Source: Gerrard and Burch, Introduction to Matrix Methods in Optics (1975), Chapter I.

The chapter develops multiplication, determinants, inversion, diagonalization, eigenvalues, and eigenvectors through examples, but it has no numbered illustrative-problem section. The essential calculation used later is summarized here so the chapter remains visible in the book sequence.

Unimodular two-by-two matrix

For

\[\begin{split}M=\begin{bmatrix}A&B\\C&D\end{bmatrix}, \qquad AD-BC=1,\end{split}\]

the characteristic polynomial is

\[\lambda^2-(A+D)\lambda+1=0.\]

Thus the eigenvalues occur as the reciprocal pair

\[\boxed{\lambda_\pm= \frac{A+D}{2}\pm \sqrt{\left(\frac{A+D}{2}\right)^2-1}}, \qquad \lambda_+\lambda_-=1.\]

An eigenvector may be scaled to \((q,1)^T\). Substitution gives

\[Cq^2+(D-A)q-B=0, \qquad q=\frac{A-D\pm\sqrt{(A+D)^2-4}}{2C}.\]

The discriminant immediately separates elliptic, hyperbolic, and parabolic matrix behavior. This is the algebraic origin of the stable, unstable, and marginal resonator classifications used in Chapter III.

Check

Substituting either root into the quadratic must give zero, and multiplying the two eigenvalues must give the determinant, unity. When \(M=I\), the double eigenvalue is \(1\), as expected.