2 -6 2 Let A=13-4 act on C2. Find the eigenvalues and a basis for each eigenspac

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2 -6 2 Let A=13-4 act on C2. Find the eigenvalues and a basis for each eigenspace in C. NOTE: The question is made up of four parts,from part (a) to (d). Please make sure you fill out all the boxes in each part before submitting it, otherwise you will use up a submission without getting any points. However, you may submit each part one at a time. (a) Eigenvalue with positive imaginary part (b) Find a basis {v) for the eigenspace corresponding to 1 . Put the real parts of v in the left column of the 2 by 2 grid below and the imaginary parts in the right column. For example, type v = 1 0 2 3 as v = 2+3-i (c) Eigenvalue with negative imaginary part = 2 = (d) Find a basis {w} for the eigenspace corresponding to 2 . Format your answer as part (b), with the real parts of w in the left column of the 2 by 2 grid below and the imaginary parts in the right column. w= symbolic formatting help

Explanation / Answer

(a) The characteristic equation of A is det(A-I2)= 0 or, 2 +2+10 = 0 or,[-(-1+3i)[-(-1-3i)] = 0.

Thus, the eigenvalues of A are 1 = -1+3i and 2= -1-3i. The eigenvalue of A with positive imaginary part is 1 = -1+3i.

(b) The eigenvector of A corresponding to the eigenvalue 1 is solutions to the equation(A- 1I2)X = 0. The solution is v1 = (1+i,1)T. Thus, a basis{v} for the eigenspace of A corresponding to the eigenvalue 1 is {(1+i,1)T}. The required matrix is v =

1

1

1

0

(c ) The eigenvalue of A with positive imaginary part is 2 = -1-3i.

(d) The eigenvector of A corresponding to the eigenvalue 2 is solutions to the equation(A- 2I2)X = 0. The solution is v2 = (1-i,1)T. Thus, a basis{w} for the eigenspace of A corresponding to the eigenvalue 2 is {(1-i,1)T}. The required matrix is w =

1

-1

1

0

1

1

1

0

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