HP HP-15C - Scalar Operations

Section 4: Using Matrix Operations 149
When A is real and symmetric (A = AT) its eigenvalues Ay are all
real and possess orthogonal eigenvectors qy. Then
and
/
if j ^ k
if; = k.
The eigenvectors (qi,q2, •••) constitute the columns of an orthogonal
matrix Q which satisfies.
Q7'AQ = diag(A1,A2,...)
and
An orthogonal change of variables x = Qz, which is equivalent to
rotating the coordinate axes, changes the equation of a family of
quadratic surfaces (x^Ax = constant) into the form
k
^~ \T(QTAQ)z = / , A,Z; — constant.
;
With the equation in this form, you can recognize what kind of
surfaces these are (ellipsoids, hyperboloids, paraboloids, cones,
cylinders, planes) because the surface's semi-axes lie along the new
coordinate axes.
The program below starts with a given matrix A that is assumed to
be symmetric (if it isn't, it is replaced by (A + AT)/2, which is
symmetric).
Given a symmetric matrix A, the program constructs a skew-
symmetric matrix (that is, one for which B = -Br) using the
formula
J tan(1/4tan"1(2ay/(an — a;y))) if i ^j and ay- ^ 0
11 \ ifi
Then Q = 2(1 + B)'1 — I must be an orthogonal matrix whose
columns approximate the eigenvalues of A; the smaller are all the
elements of B, the better the approximation. Therefore QTAQ must
be more nearly diagonal than A but with the same eigenvalues. If