Thread: Diagonilizablity 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.

Originally Posted by Chris11

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.

eigenvalue of of multiplicity is the charateristic pol. of , and thus is diagonalizable is the minimal pol. of .

Tonio

what is minimal pol?

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.