{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

week13tutsols

# week13tutsols - THE UNIVERSITY OF SYDNEY Math2968...

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}