 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1/6/v/solution

Collect the paired orthonormal bases from part (iv), followed by an orthonormal basis of

ker M^{2} = ker M

Q

of these columns is an orthogonal matrix, and

Q^{T} MQ = j = 1 ⨁ r (0 μ_{j} - μ_{j} 0) \oplus 0_{n - 2 r}, μ_{j} > 0.

Indeed, each pair obeys

M e_{j} = μ_{j} f_{j}

M f_{j} = - μ_{j} e_{j}

, and all kernel vectors are annihilated. This proves the real block form of a skew-symmetric matrix directly, including repeated block parameters. In particular its rank is

2 r

, always even.

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