Thread: 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).

Originally Posted by chadlyter

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 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.

Originally Posted by ThePerfectHacker

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.

Hence , and are roots of , and any diagonal matrix with each diagonal element equal to a root (not necessarily distinct) satisfies the charateristic equation.

RonL