Let A be an nxn matrix with eigenvalue ∆ with multiplicity n. Show that A is diagonizable iff A=∆I, where I is the nxn idenity.

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- Apr 26th 2010, 08:51 AMChris11Diagonilizablity question
Let A be an nxn matrix with eigenvalue ∆ with multiplicity n. Show that A is diagonizable iff A=∆I, where I is the nxn idenity.

- Apr 26th 2010, 09:07 AMtonio

$\displaystyle \Delta$ eigenvalue of $\displaystyle A$ of multiplicity $\displaystyle n\iff (x-\Delta)^n$ is the charateristic pol. of $\displaystyle A$ , and thus $\displaystyle A$ is diagonalizable $\displaystyle \iff x-\Delta$ is the minimal pol. of $\displaystyle A\iff A=\Delta I $ .

Tonio - Apr 26th 2010, 10:38 AMChris11
what is minimal pol?

- Apr 27th 2010, 02:20 AMHallsofIvy
The polynomial, p(x), of minimum degree such that p(A)v= 0 for all v in the vector space. Since every matrix satisfies its own characteristic equation, the minimal polynomial for a matrix must be a factor of the characteristic polynomial.