•This is the secular equation (Lat. saeculum = age, generation,
century). It first arose in
astronomical calculations about the long-term behavior of planetary orbits. The meaning “non-religious” arose
later, from “temporary”.
• The secular
equation can be compactly written by using a matrix notation:
•HC=ESC
•(Explain matrix,
vector notation).
This is a generalized
matrix eigenvalue problem. If we omit S then it is HC=EC a
matrix
eigenvalue problem. We can omit S if Sii=1, Sij=0 if (i¹j). Sij=dij; dij=Kronecker delta =1 if i=j, =0 if i¹j.
• Notice that Sij=dij if the basis functions {c} are normalized and orthogonal to each other, with one word orthonormal. Orthonormality greatly
simplifies things.
•Mathematicians have developed efficient algorithms for solving both kinds of matrix
eigenvalue problem.