Thursday, May 1, 2014

Rotation of quadratic forms

Rotation of quadratic forms

When A is a multiple of I, then the quadratic form $x^TAx$ is a circle at a give $f(x)$ level. When there is a rotation, $x = Ru$, then $f(u) = u^T R^T A R u$.

Notice that $A = \alpha I$, and $R^TAR = \alpha R^TR = \alpha I = A$, and $f(u) = u^TAu$, i.e. the rotation has no effect on the quadratic form.

Also note that when A is a multiple of I, R can be put before or after A:

\begin{align} R^TAR &= A \\ AR &= RA \\ \end{align}