Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / 6 / i / b

Table of contents
- Solution b

Solution

For any Hermitian matrix

K

x^{†} K x

is real because its complex conjugate equals

x^{†} K^{†} x = x^{†} K x

. Applying this to

H, S

in part (a) gives

Re λ = x^{†} H x, Im λ = x^{†} S x .

These are the real and imaginary quadratic forms associated with the numerical range of

A

.

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