In this question, you will build a matrix from the eigenvectors and eigenvalues,

Question

In this question, you will build a matrix from the eigenvectors and eigenvalues, instead of the other way around. The matrix A has three eigenvalues: lambda_1 = -10 with eigenvector u_1 = (1, 1, 0), lambda_2 = 9 with eigenvector u_2 = (1, -1, 1) and lambda_3 = 6 with eigenvector u_3 = (l, -1, - 2). Normalize the eigenvectors u_i to give v_i. Enter them in the usual format e.g. [1, 2, 3]. v_1 = ______ v_2 = ______ v_3 = ______ Recall that you can build an orthogonal matrix P whose columns are that set of orthonormal eigenvectors. Then P^TAP = D, where D is a diagonal matrix that contains the eigenvalues along the diagonal. We now have P and D, so can find A from the formula A = PDP^T. Enter the matrix A as a list of row vectors. ________

Explanation / Answer

              It may be observed that v1,v2,v3 are orthonormal vectors.

ALTERNATIVELY

(a) We know that, when the eigenvectors of A are distinct and linearly independent, as in this case, then, as per the diagonalization theorem, the matrix A can be expressed as A = PDP-1 , where P is the matrix whose columns are the eigenvectors of A and D is the diagonal matrix with the eigenvalues of A ( in the same order)as elements on its leading diagonal. Then P =

1

1

1

1

-1

-1

0

1

-2

and D =

-10

0

0

0

9

0

0

0

6

Hence A = PDP-1 =

-1

-9

1

-9

-1

-1

1

-1

7

1

1

1

1

-1

-1

0

1

-2

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