Paper 3, Section II, D

Algebra and Geometry | Part IA, 2007

Let AA be a real symmetric n×nn \times n matrix. Prove that every eigenvalue of AA is real, and that eigenvectors corresponding to distinct eigenvalues are orthogonal. Indicate clearly where in your argument you have used the fact that AA is real.

What does it mean to say that a real n×nn \times n matrix PP is orthogonal ? Show that if PP is orthogonal and AA is as above then P1APP^{-1} A P is symmetric. If PP is any real invertible matrix, must P1APP^{-1} A P be symmetric? Justify your answer.

Give, with justification, real 2×22 \times 2 matrices B,C,D,EB, C, D, E with the following properties:

(i) BB has no real eigenvalues;

(ii) CC is not diagonalisable over C\mathbb{C};

(iii) DD is diagonalisable over C\mathbb{C}, but not over R\mathbb{R};

(iv) EE is diagonalisable over R\mathbb{R}, but does not have an orthonormal basis of eigenvectors.

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