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 / i

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- Solution i

Solution

The given symmetric matrices satisfy

E_{j}^{T} E_{j} = E_{j}^{2} = I

, so both are orthogonal matrices. Put

J = E_{1} E_{2}

,

c = cos (θ /2)

and

s = sin (θ /2)

. Anticommutation gives

J^{T} = E_{2} E_{1} = - J, J^{2} = E_{1} E_{2} E_{1} E_{2} = - I .

Thus

A = c I + s J

and

A^{T} = c I - s J

A A^{T} = (c^{2} + s^{2}) I = I

. These are orthogonal rotations generated by anticommuting involutions. Also

A - A^{T} = 2 s J

. Since

J^{2} = - I

makes

J

invertible, $A$ is symmetric exactly when $sin (θ /2) = 0$ .

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