Ch. 8 Diagonalization Flashcards
1
Q
Definition of 2 matrices being similar
A
A and B are similar if there exists an invertible matrix s.t A = M^-1BM
(have the same eigenvalues)
2
Q
How to show that a matrix A is similar to a diagonal matrix
A
A = MDM^-1
where D is a matrix with the eigenvalues of A along the diagonal and 0s everywhere else
M is the matrix made up of all the eigenvectors of A
3
Q
Cayley-Hamilton Theorem
A
If A is a square matrix with characteristic polynomial p(t). Replace the t with the matrix A to get p(A) Then p(A) = 0
4
Q
Conditions for matrix A being diagonalisable
A
If A has distinct eigenvalues, A is diagonalisable
5
Q
Given a matrix A, how to find e.g. A^6
A
A^6 = M(D^6)(M^-1)
6
Q
How to diagonalise a matrix
A
D = (M^-1)AM
where M is the eigenvectors of A in a matrix
7
Q
Given A, how to find a matrix B such that e.g. B^3 = A
A
A = M(D^3)M^-1 B = MDM^-1
