This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 Tinvariant 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 Tinvariant. 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 1dimensional. 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...
View
Full
Document
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.
 One '09
 easdown
 Algebra

Click to edit the document details