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week13tutsols - THE UNIVERSITY OF SYDNEY Math2968...

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THE UNIVERSITY OF SYDNEY Math2968 Algebra (Advanced) Semester 2 Tutorial Solutions Week 13 2008 1. (for general discussion) Why is a generalised eigenspace invariant with respect to its operator? Solution It is sufficient to verify that W is T -invariant where T : V V is a linear operator and W = ker( T λ id) k for some positive integer k and scalar λ . If v W then, since T commutes with any polynomial expression of T in the arithmetic of operators (where multiplication is composition), ( T λ id) k ( T ( v )) = bracketleftbig ( T λ id) k T bracketrightbig ( v ) = bracketleftbig T ( T λ id) k bracketrightbig ( v ) = T bracketleftbig ( T λ id) k ( v ) bracketrightbig = T (0) = 0 . This proves T ( W ) W . In particular, any generalised eigenspace of T is T -invariant. 2. Write down general forms for all 4 × 4 Jordan matrices where the sizes of Jordan blocks are nonde- creasing down the diagonal. Solution λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 0 0 0 0 λ 4 λ 1 0 0 0 0 λ 2 0 0 0 0 λ 3 1 0 0 0 λ 3 λ 1 1 0 0 0 λ 1 0 0 0 0 λ 2 1 0 0 0 λ 2 λ 1 0 0 0 0 λ 2 1 0 0 0 λ 2 1 0 0 0 λ 2 λ 1 0 0 0 λ 1 0 0 0 λ 1 0 0 0 λ 3. Suppose T : V V is an operator, dim V = 3 and the distinct eigenvalues of T are λ 1 and λ 2 , both of whose corresponding eigenspaces are 1-dimensional. Describe the possible Jordan forms for T . Solution λ 1 0 0 0 λ 2 1 0 0 λ 2 λ 2 1 0 0 λ 2 0 0 0 λ 1 λ 2 0 0 0 λ 1 1 0 0 λ 1 λ 1 1 0 0 λ 1 0 0 0 λ 2 4. Find e tA where A is each of the following matrices: (a) bracketleftbigg 1 0 0 2 bracketrightbigg (b) bracketleftbigg 2 1 1 2 bracketrightbigg (c) bracketleftbigg 5 6 3 4 bracketrightbigg (d) 1 0 0 1 1 0 0 1 1 Solution (a) bracketleftbigg e t 0 0 e 2 t bracketrightbigg (b) bracketleftbigg e 2 t cos t e 2 t sin t e 2 t sin t e 2 t cos t bracketrightbigg (c) bracketleftbigg e t + 2 e 2 t 2 e t 2 e 2 t e t + e 2 t 2 e t e 2 t
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