# Thread: Diagonal Matrices from Characteristic Poly

1. ## Diagonal 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).

The charachteristic polynomial of two similar matrices are the same. Thus, if $A$ has the charachteristic polynomial $X^3 - 4X^2 + 5X - 2$ then $A^3 - 4A^2 + 5A - 2I = \bold{0}$ 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 $A$ is a symettric matrix it means it is diagnolizable and the rest follows.
Thus, you need to find $\left[ \begin{array}{ccc}a&0&0\\0&b&0\\0&0&c \end{array} \right]$ which solve this polynomial equation.
The charachteristic polynomial of two similar matrices are the same. Thus, if $A$ has the charachteristic polynomial $X^3 - 4X^2 + 5X - 2$ then $A^3 - 4A^2 + 5A - 2I = \bold{0}$ 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 $A$ is a symettric matrix it means it is diagnolizable and the rest follows.
Thus, you need to find $\left[ \begin{array}{ccc}a&0&0\\0&b&0\\0&0&c \end{array} \right]$ which solve this polynomial equation.
Hence $a$, $b$ and $c$ are roots of $x^3 - 4x^2 + 5x - 2 = {0}$, and any diagonal matrix with each diagonal element equal to a root (not necessarily distinct) satisfies the charateristic equation.