Paper 1, Section II, B

Vectors and Matrices | Part IA, 2018

Let AA be a real symmetric n×nn \times n matrix.

(a) Prove the following:

(i) Each eigenvalue of AA is real and there is a corresponding real eigenvector.

(ii) Eigenvectors corresponding to different eigenvalues are orthogonal.

(iii) If there are nn distinct eigenvalues then the matrix is diagonalisable.

Assuming that AA has nn distinct eigenvalues, explain briefly how to choose (up to an arbitrary scalar factor) the vector vv such that vTAvvTv\frac{v^{T} A v}{v^{T} v} is maximised.

(b) A scalar λ\lambda and a non-zero vector vv such that

Av=λBvA v=\lambda B v

are called, for a specified n×nn \times n matrix BB, respectively a generalised eigenvalue and a generalised eigenvector of AA.

Assume the matrix BB is real, symmetric and positive definite (i.e. (u)TBu>0\left(u^{*}\right)^{T} B u>0 for all non-zero complex vectors uu ).

Prove the following:

(i) If λ\lambda is a generalised eigenvalue of AA then it is a root of det(AλB)=0\operatorname{det}(A-\lambda B)=0.

(ii) Each generalised eigenvalue of AA is real and there is a corresponding real generalised eigenvector.

(iii) Two generalised eigenvectors u,vu, v, corresponding to different generalised eigenvalues, are orthogonal in the sense that uTBv=0u^{T} B v=0.

(c) Find, up to an arbitrary scalar factor, the vector vv such that the value of F(v)=vTAvvTBvF(v)=\frac{v^{T} A v}{v^{T} B v} is maximised, and the corresponding value of F(v)F(v), where

A=(4202300010) and B=(210110003)A=\left(\begin{array}{ccc} 4 & 2 & 0 \\ 2 & 3 & 0 \\ 0 & 0 & 10 \end{array}\right) \quad \text { and } \quad B=\left(\begin{array}{ccc} 2 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 3 \end{array}\right)

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