Domain Range Domain Range Domain Range Domain Range

Domain Range

Domain Range

Domain Range

Domain Range

definition: T is a linear transformation , EIGENVECTOR EIGENVALUE

A is the matrix for a linear transformation T relative to the STANDARD BASIS

T T

The matrix for T relative to the basis T T

The matrix for T relative to the basis Diagonal matrix Eigenvectors for T T T

The matrix for a linear transformation T relative to a basis of eigenvectors will be diagonal

To find eigenvalues and eigenvectors for a given matrix A: Solve for and A = I =( I - A)

To find eigenvalues and eigenvectors for a given matrix A: Solve for and Remember: is a NONZERO vector in the null space of the matrix: ( =( I I - - A) A)

is a NONZERO vector in the null space of the matrix: ( The matrix ( I det I - A) A ) has a nonzero vector in its null space iff: ( I - A) = 0

det ( I - A) =

det ( I - A) =

det ( I - A) = det This is called the characteristic polynomial

det ( I - A) = det =0 the eigenvalues are 2 and -1

( 2 I - A) = the eigenvectors belonging to 2 are nonzero vectors in the null space of 2 I - A =

( -1 I - A ) = the eigenvectors belonging to -1 are nonzero vectors in the null space of -1 I - A =

Matrix for T relative to standard basis

Matrix for T relative to columns of P

Basis of eigenvectors

eigenvalues