Eigenvectors/values Flashcards
(16 cards)
The eigenvalues of a triangular matrix are
The entries on its diagonal
What is an eigenspace
The set of all vectors scaled by an eigenvector. Each LI eigenvector will have its own eigenspace
Whats the characteristic equation
det(A-eval*i)=0
The set of eigenvectors with distinct eigenvalues is
A LI set
Sum of eigenvalues =
Trace(A), which is the sum of the diagonal entries of A
The product of all the eigenvalues is
The det(A)
What does it mean for A to be similar to B
There exists an invertible matrix P such that A=PBP^-1
What does A~B mean
They are similar
If A~B, then they have the same
Characteristic polynomial and same eigenvalues with same multiplicity
Eigenvector equation
Ax=lambda*x, x is evect, lambda is eval
If A~B and x is an evect of A wrt lambda, then
P^-1x is the evect of B corresponding to lambda
If u have a matrix that is just a diagonal, what trick can u do to multiply it by a matrix to the right?
Multiply each element in the right matrix row with the corresponding pivot on the left
A matrix A is diagonalizable if
There exists a similar matrix to it which is a diagonal matrix
A non matrix is diagonalizable if
It has n LI eigenvectors, the columns of P are thr eigenvectors of A and the diagonal entries of D are the corresponding eigenvalues (A=PDP^-1)
If A has n distinct evals then
A is automatically diagonalizable and the events are LI, forming the columns of P
If some evals of A are repeated then can A still be diagonalizable?
Yes, but only if the repeated evals can provide as many LI evects as their multiplicity
