A 3 x 3 symmetric matrix A has characteristic polynomial (~ -1)^2(~ - 2). Find all diagonal matrices similar to A. Any ideas? Never seen a question like this before. When multiplied out the poly is (~^3 - 4~^2 + 5~ - 2).

Printable View

- July 7th 2008, 11:13 AMchadlyterDiagonal Matrices from Characteristic Poly
A 3 x 3 symmetric matrix A has characteristic polynomial (~ -1)^2(~ - 2). Find all diagonal matrices similar to A. Any ideas? Never seen a question like this before. When multiplied out the poly is (~^3 - 4~^2 + 5~ - 2).

- July 7th 2008, 12:38 PMThePerfectHacker
The charachteristic polynomial of two similar matrices are the same. Thus, if has the charachteristic polynomial then by the Cayley-Hamilton theorem. Of course, this theorem is very advanced and you probably never seen it before. Therefore, there is a weaker version for this theorem which states that a diagnolizable matrix satisfies its charachteristic polynomial. Since is a symettric matrix it means it is diagnolizable and the rest follows.

Thus, you need to find which solve this polynomial equation. - July 7th 2008, 12:57 PMCaptainBlack