Paper 1, Section II, 6B\mathbf{6 B}

Vectors and Matrices | Part IA, 2016

The n×nn \times n real symmetric matrix MM has eigenvectors of unit length e1,e2,,en\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}, with corresponding eigenvalues λ1,λ2,,λn\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}, where λ1>λ2>>λn\lambda_{1}>\lambda_{2}>\cdots>\lambda_{n}. Prove that the eigenvalues are real and that eaeb=δab\mathbf{e}_{a} \cdot \mathbf{e}_{b}=\delta_{a b}.

Let x\mathbf{x} be any (real) unit vector. Show that

xTMxλ1\mathbf{x}^{\mathrm{T}} M \mathrm{x} \leqslant \lambda_{1}

What can be said about x\mathbf{x} if xTMx=λ1?\mathbf{x}^{\mathrm{T}} M \mathbf{x}=\lambda_{1} ?

Let SS be the set of all (real) unit vectors of the form

x=(0,x2,,xn)\mathbf{x}=\left(0, x_{2}, \ldots, x_{n}\right)

Show that α1e1+α2e2S\alpha_{1} \mathbf{e}_{1}+\alpha_{2} \mathbf{e}_{2} \in S for some α1,α2R\alpha_{1}, \alpha_{2} \in \mathbb{R}. Deduce that

MaxxSxTMxλ2\underset{\mathbf{x} \in S}{\operatorname{Max}} \mathbf{x}^{\mathrm{T}} M \mathbf{x} \geqslant \lambda_{2}

The (n1)×(n1)(n-1) \times(n-1) matrix AA is obtained by removing the first row and the first column of MM. Let μ\mu be the greatest eigenvalue of AA. Show that

λ1μλ2\lambda_{1} \geqslant \mu \geqslant \lambda_{2}

Typos? Please submit corrections to this page on GitHub.