CLA
AB nxn, does AB and BA have the same eigenvalues?
Yes, ABv=lambda v then BA(Bv)=lambda Bv
A,B mxn, AB and BA same eigenvalues?
No, the matrices change size and therefore the number of eigenvalues.
A,B nxn are symmetric do AB have BA have the same eigenvalues? real eigenvalue???
No, since you can find a counter example where AB is non - symmetric
Do symmetric matrices have real eigenvalues?
Yes
Is A=A^T a symmetric matrix?
Yes
Definition of a positive definite matrice M? and the properties
z^TMz > 0 for any column vector z. Square and symmetric and all the eigenvalues are positive. (not necessarly symmetric)
When is a matrix A invertiable?
If it has full column rank, nonzero determinante and square.
What is the Cholesky factorization? and when does it work?
B=LL^T, where L is a lowertraingular matrix, with real and positive diagonal entries. Possible id B is possitvie semi definite. For a hermitian matrix A, A=LL*.
What is an hermitian matrix?
A=A^H, where H denotes the complex conjugate and transpose.
What is the Schur decomposition of A?
A=QUQ^-1, where Q is a unitary matrix such rhat the inverse of Q is also the conjugate transpose. U is an uppertriangular matrix.
What is a unitary matrix?
A matrix is unitary if its inverse equals its conjugate transpose.
what is the angle formula for vectors?
cos(theta)=(x^Ty)/(||x||||y||)
What is the power method?
x_k = Ax_k/||Ax_k||
which eigenvalue does the power mothod converge towards, and why?
The largest eigenvalue, since it is the dominant term in the linear combination. lambda_1^p grows fast then the others, where p is the number of iterations.