Algebra 2C Flashcards
What does the spectral theorem state
Spectral theorem states that :
Linear operator phi is orthogonally diag iff phi is normal (field =C)
If V is an I.P space and linear operator phi is self-adjoint, then what are properties of phi
If V is an I.P space and linear operator phi is self-adjoint, then what are properties of phi are:
Any eigenvalue of phi is real
If v,w in V are eigenvalues then v is perp to w I.e inner(v,w) = 0
If V is a real I.P and phi is self adjoint, what does phi have
If V is a real I.P and phi is self adjoint then phi has a real eigenvalue
What is spectral thm for symmetric and hermitan matrices
Spectral thm for symmetric matrices is:
If matrix A is symmetric, then E orthogonal matrix P s.t P^-1AP is diagonal
If A is a hermitian, then E unitary matrix P s.t P^-1AP is diag
What is the singular value decomposition
Singular value decomposition is:
If phi is self adjoint this implies E an orthonormal eigenbasis implying that for all v in V can be written as a linear combination.
What is singular value decomposition for matrices
Singular value decomposition for matrices is:
E unitary matrix P s.t P^-1AP= diagonal matrix
What are singular values of phi
Singular values of phi are sigma1 … sigma n where sigma i = root(mu i) and mu i are eigenvalues of phi* o phi
What is singular value decomposition thm
Singular value decomposition thm is:
If phi has singular values sigma1 … sigma n. Then E orth.normal bases u1,…un and w1,…,wn of C such that phi(v) = sum 1 to n sigma i (inner product of ui,v) wi
How can any n x n matrix be written
Any matrix can be written by a 1 x n matrix multiplied by an n x 1 matrix
What does spectral thm state for real self-adjoint operators in V real I.P space
For real self-adjoint operators, spectral thm states :
Phi is orthogonally diagonalisable iff phi is self-adjoint
