Question

4. Working within an eigenbasis for A. 1 2i A = -2i 1 a) Given matrix A, solve for the eigenbasis, {le), le2)} (remember: le;) are column vectors). Note: I expect you to pull out a common factor so that the first entry in the vectors is positive and real. b) Solve for the projection matrices: le) (e;]-. 2 c) Explicitly show the result of the operation:le) (el. i-1 d) Explicitly show the result of the operation: , ; ei) (e;| (where Ai are the eigenvalues of i-1 A). e) Extrapolate from (c) and (d) to make a general statement about m Σλ) e le) (el and lea) (e for any dimensional matrix A.

Answer 1

Here we 1st calculate the eign values of the given matrix and the find out the corresponding eign vectors , then we calculte the projection matrix respectively , Here I am attaching the solution below,

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HE801 (6) Wo have got leay 27 (1) here the projection matrixes are leitheiz (:) (12) and leap kesle (2) C1-) con lei)<it - lei) <0,7 + 15754 69 Š xi lei><cil - |41}<47 +1 ks) (el.