Suppose that A is a square matrix and it has two distinct eigenvalues, lamda 1 and lamda 2. and that dim(E_lamda1)=n-1. Prove that A is diagonalizable.
Suppose that A is a square matrix and it has two distinct eigenvalues, lamda 1 and lamda 2. and that dim(E_lamda1)=n-1. Prove that A is diagonalizable.
GIVEN INFORMATION:
A is a square matrix ( n x n ).
A has two distinct eigenvalues, and .
Proof:
For an n x n matrix, the dimensions of the eigenspaces must add up to n for the matrix to be diagonalizable. We are given . Furthermore, we know that and . Thus, .
So, finally, . We have proven that A is diagonalizable.
-Andy