Paper 1, Section II, B

Vectors and Matrices | Part IA, 2017

(a) Show that a square matrix AA is anti-symmetric if and only if xTAx=0\mathbf{x}^{T} A \mathbf{x}=0 for every vector x\mathbf{x}.

(b) Let AA be a real anti-symmetric n×nn \times n matrix. Show that the eigenvalues of AA are imaginary or zero, and that the eigenvectors corresponding to distinct eigenvalues are orthogonal (in the sense that xy=0\mathbf{x}^{\dagger} \mathbf{y}=0, where the dagger denotes the hermitian conjugate).

(c) Let AA be a non-zero real 3×33 \times 3 anti-symmetric matrix. Show that there is a real non-zero vector a such that Aa=0A \mathbf{a}=\mathbf{0}.

Now let b\mathbf{b} be a real vector orthogonal to a\mathbf{a}. Show that A2b=θ2bA^{2} \mathbf{b}=-\theta^{2} \mathbf{b} for some real number θ\theta.

The matrix eAe^{A} is defined by the exponential series I+A+12!A2+I+A+\frac{1}{2 !} A^{2}+\cdots Express eAae^{A} \mathbf{a} and eAbe^{A} \mathbf{b} in terms of a,b,Ab\mathbf{a}, \mathbf{b}, A \mathbf{b} and θ\theta.

[You are not required to consider issues of convergence.]

Typos? Please submit corrections to this page on GitHub.