# Diagonilizablity question

• Apr 26th 2010, 08:51 AM
Chris11
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.
• Apr 26th 2010, 09:07 AM
tonio
Quote:

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.

\$\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 AM
Chris11
what is minimal pol?
• Apr 27th 2010, 02:20 AM
HallsofIvy
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.