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# FP2 Matrices Question watch

1. The matrix A has eigenvectors ( 3 2 1 ), ( -7 2 3), (1 5 4)
with corresponding eigenvalues -1,1,0
Express A as the product of three matrices and hence find A.

Unsure how to approach this question, any help would be appreciated!
2. I think you're supposed to know that D = P^T x A x P

where
D is a matrix with the eigenvalues on the leading diagonal and zero elsewhere
P is a matrix made up of the normalised eigenvectors (in columns) in the same order as D
P^T is the transpose of P

You can rearrange for A by judicious pre or post-multiplication by matrix inverses.

Caveat: there might be a better way, but I don't know of one off-hand.
3. (Original post by old_engineer)
I think you're supposed to know that D = P^T x A x P

where
D is a matrix with the eigenvalues on the leading diagonal and zero elsewhere
P is a matrix made up of the normalised eigenvectors (in columns) in the same order as D
P^T is the transpose of P

You can rearrange for A by judicious pre or post-multiplication by matrix inverses.

Caveat: there might be a better way, but I don't know of one off-hand.
You need P inverse not P transpose (if your eigenvectors are ortho-normal they are the same matrix but this is not true in general).
4. Ah that makes sense- thank you!
5. (Original post by DFranklin)
You need P inverse not P transpose (if your eigenvectors are ortho-normal they are the same matrix but this is not true in general).
Yes, thank you for the correction. I had carelessly posted a special case that I think is not applicable to the eigenvectors in this question.

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