{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# solution9 - which is represented by the matrix A in the...

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

In each of the following cases, let T be the linear operator on R 2 which is represented by the matrix A in the standard ordered basis for R 2 , and let U be the linear operator on C 2 represented by A in the standard ordered basis. Find the characteristic polynomial for T and that for U , find the characteristic values of each operator, and for each such characteristic value c find a basis for the corresponding space of characteristic vectors. A = 1 0 0 0 A = 2 3 - 1 1 A = 1 1 1 1 In all cases, denote by B c the basis for the subspace corresponding to the characteristic value c First matrix The characteristic polynomial is x ( x - 1) . The roots are 0 and 1 . B 0 = { (0 , 1) } and B 1 = { (1 , 0) } . In this case the real and complex cases are the same. Second matrix The characteristic polynomial is 5 - 3 x + x 2 which has no real roots. The complex eigenvalues are 3 2 + i 1 2 11 , 3 2 - i 1 2 11 , B 3 2 + i 1 2 11 = { (1 , - 1 6 + i 1 6 11) } and B 3 2 - i 1 2 11 = { (1 , - 1 6 - i 1 6 11) } Third matrix The characteristic polynomial is x 2 - 2 x . The roots are 0 and 2 . B 0 = { ( - 1 , 1) } and B

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 ]}

### Page1 / 3

solution9 - which is represented by the matrix A in the...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online