Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 5

Since $M^T=-M$, $(M^2{)}^T=(M^T{)}^2=M^2$, so $M^2$ is a real symmetric matrix. Moreover,proving that it is a negative semidefinite matrix.

For odd $n$, determinant invariance under transpose givesso $\det M=0$. Hence $M$ and $M^2$ have a zero eigenvalue. Finally, if $M^{2}w=0$, then $0=w^T{M}^{2}w=-\Vert Mw{\Vert }^2$, so $Mw=0$. Thuswhere equality of these kernels includes the reverse implication, which is immediate.

Because $M$ commutes with $M^2$,Its squared Euclidean norm isA scalar equals its transpose, while $M^T=-M$, so $u_1^{T}M{u}_1=-u_1^{T}M{u}_1=0$. ConsequentlyThus each negative eigenvalue has at least this pair of orthonormal eigenvectors.

The plane spanned by $u_1,v_1$ is an invariant subspace for $M$, sinceIts orthogonal complement is also invariant: for $w$ orthogonal to both, $u_1^{T}Mw=-(M{u}_1{)}^{T}w=0$ and $v_1^{T}Mw=-(M{v}_1{)}^{T}w=0$. The restriction of the real symmetric matrix $M^2$ to this nonempty complement therefore has a unit eigenvector $u_2$, by the spectral theorem for real symmetric matrices.

If its eigenvalue is $-{\mu }_2^2<0$, the argument in part (b) shows that $v_2=M{u}_2/{\mu }_2$ is a unit eigenvector orthogonal to $u_2$. It also lies in the same complement; explicitly,Repeating the construction decomposes every negative eigenspace into orthogonal pairs. The zero eigenspace has dimension $n-2r$ for some number $r$ of pairs, and therefore has the same parity as $n$. This is the paired eigenspaces of the square of a real skew-symmetric matrix property.

For $n=2m$, there can be $m$ distinct negative eigenvalues and no zero, or at most $m-1$ negative ones plus zero, because a nonempty zero eigenspace has dimension at least two. For $n=2m+1$, there are at most $m$ distinct negative ones plus the mandatory zero. HenceThe bound is attained by skew two-by-two blocks with distinct positive parameters, adding one zero direction when $n$ is odd.

The required actions have already been identified:In three dimensions choose a unit vector $w$ perpendicular to both. Its one-dimensional orthogonal complement to the pair is invariant under $M$. On a one-dimensional real space, a skew-symmetric matrix must act as zero: $Mw=\eta w$ would imply $\eta =w^{T}Mw=0$. Thus $Mw=0$.

The columns of $W=(u_1{v}_{1}w)$ form an orthonormal basis, so $W$ is an orthogonal matrix. The displayed actions giveand thereforeIf $M=0$, take ${\mu }_1=0$ and any orthogonal matrix $W$. Otherwise its nonzero two-dimensional block supplies the pair used above.