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).

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- Jul 7th 2008, 10:13 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).

- Jul 7th 2008, 11:38 AMThePerfectHacker
The charachteristic polynomial of two similar matrices are the same. Thus, if $\displaystyle A$ has the charachteristic polynomial $\displaystyle X^3 - 4X^2 + 5X - 2$ then $\displaystyle A^3 - 4A^2 + 5A - 2I = \bold{0}$ 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 $\displaystyle A$ is a symettric matrix it means it is diagnolizable and the rest follows.

Thus, you need to find $\displaystyle \left[ \begin{array}{ccc}a&0&0\\0&b&0\\0&0&c \end{array} \right]$ which solve this polynomial equation. - Jul 7th 2008, 11:57 AMCaptainBlack
Hence $\displaystyle a$,$\displaystyle b$ and $\displaystyle c$ are roots of $\displaystyle x^3 - 4x^2 + 5x - 2 = {0}$, and any diagonal matrix with each diagonal element equal to a root (not necessarily distinct) satisfies the charateristic equation.

RonL