What do eigenvalues and eigenvectors of a Matrix A tell us about a transformation T (T: R^3 -> R^3)?
I want to know if I'm on the right page, close to, or not correct about any of these. Let's say I have a 3x3 matrix and its eigenvalues are 0,2, and 5.
1) Eigenvalues can tell us if A is an invertible matrix if and only if zero is not an eigenvalue of A, but because it is in this example we know that A is not an invertible matrix. That means that the transformation of A is not invertible as well.
2) If the vectors are eigenvectors that correspond with the eigenvalues of an nxn matrix A then the set of vectors are linearly independent. Since the eigenvalues of A and eigenvectors correspond to each other then the set of vectors is linearly independent. Meaning the vectors of the transformation of A are linearly independent.
3) If A has 'n' linearly independent eigenvectors then the nxn matrix A is diagonalizable. Because A has all 3 linearly independent eigenvectors we can conclude that A is diagonalizable and the Transformation of A is diagonalizable.