Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 3 / b / ii

Table of contents
- Solution ii

Solution

Use

J E_{1} = - E_{2}

,

E_{1} J = E_{2}

and

J E_{1} J = E_{1}

to expand the matrix product:

A E_{1} A^{T} = c^{2} E_{1} + cs (J E_{1} - E_{1} J) - s^{2} J E_{1} J = (c^{2} - s^{2}) E_{1} - 2 cs E_{2} .

The double-angle identities therefore give

A E_{1} A^{T} = cos θ E_{1} - sin θ E_{2} .

Similarly

J E_{2} = E_{1}

,

E_{2} J = - E_{1}

and

J E_{2} J = E_{2}

, so

A E_{2} A^{T} = sin θ E_{1} + cos θ E_{2} .

This conjugation rotates the two-dimensional span of the two matrices through the double angle described by orthogonal rotations generated by anticommuting involutions.

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