Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 18

The symmetric and antisymmetric parts of a matrix areIndeed $S+A=M$, while the transpose identities $(M^T{)}^T=M$ give $S^T=S$ and $A^T=-A$. Thus $S$ is a symmetric matrix and $A$ is an antisymmetric matrix. The decomposition is also unique: a matrix that is both symmetric and antisymmetric must equal its own negative and hence vanish.

Take the dot product of the orthogonal basis expansion with ${\mathbf{u}}_j$. All terms except the $j$th vanish by orthogonality, leavingThereforeThe basis is not assumed normalized, so its squared lengths must remain in the denominators. Applying $S$ and using $S{\mathbf{u}}_i={\lambda }_i{\mathbf{u}}_i$ gives

Induction on $p$ gives $S^p{\mathbf{u}}_i={\lambda }_i^p{\mathbf{u}}_i$, including $p=0$. The absolutely convergent matrix exponential series may therefore be applied term by term to each eigenvector:ThusFor a repeated eigenvalue, every nonzero vector in its eigenspace remains an eigenvector. Because the real exponential function is injective, distinct real eigenvalues of $S$ do not merge under this transformation.

Since every eigenvalue is nonzero, $S$ is an invertible matrix. Multiplying $S{\mathbf{u}}_i={\lambda }_i{\mathbf{u}}_i$ by the inverse matrix givesHence the eigenvectors are unchanged and the corresponding eigenvalues are $1/{\lambda }_i$.

Choose the unnormalized eigenvectorsTheir linear combination giving the specified vector isApplying the reciprocal eigenvalues of the inverse matrix therefore yieldsIn Cartesian coordinates, this is $(-1/3,2/3{)}^T$, and direct matrix multiplication gives $S\mathbf{y}=(1,0{)}^T$.

The defining condition for an antisymmetric matrix forcesIts dot product with $(x,y{)}^T$ is zero, and its Euclidean norm is $|a|\sqrt{x^2+y^2}$. Thus it combines a right-angle rotation with a uniform dilation by $|a|$. More precisely, the rotation is clockwise for $a>0$ and anticlockwise for $a<0$. A direction-independent description is a clockwise right-angle rotation followed by multiplication by the signed scalar $a$.

A second application givesGeometrically this is a half-turn combined with a dilation by $a^2$. Because $A$ is nonzero, $a\ne 0$.

Let $J=A/a$, so $J^2=-I$. The even and odd powers in the matrix exponential giveHere the two scalar sums are the Taylor series for cosine and sine. ConsequentlyWith the usual convention that positive angles turn anticlockwise, this rotation matrix rotates by $-a$; the parameter in the form printed in the paper is $\theta =a$.