week13tutsols - THE UNIVERSITY OF SYDNEY Math2968 Algebra...

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Unformatted text preview: 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 id) k for some positive integer k and scalar . If v W then, since T commutes with any polynomial expression of T in the arithmetic of operators (where multiplication is composition), ( T id) k ( T ( v )) = bracketleftbig ( T id) k T bracketrightbig ( v ) = bracketleftbig T ( T id) k bracketrightbig ( v ) = T bracketleftbig ( T 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 1 2 3 4 1 2 3 1 3 1 1 1 2 1 2 1 2 1 2 1 2 1 1 1 3. Suppose T : V V is an operator, dim V = 3 and the distinct eigenvalues of T are 1 and 2 , both of whose corresponding eigenspaces are 1-dimensional. Describe the possible Jordan forms for T . Solution 1 2 1 2 2 1 2 1 2 1 1 1 1 1 1 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 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...
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This note was uploaded on 09/12/2009 for the course MATH 2968 taught by Professor Easdown during the One '09 term at University of Sydney.

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week13tutsols - THE UNIVERSITY OF SYDNEY Math2968 Algebra...

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