Spectral Methods Flashcards
What is a singular matrix?
A matrix with no inverse
What is a non-singular matrix?
A matrix with an inverse
What is the result of multiplying a matrix with its inverse?
An identity matrix
What is the determinant of matrix A if A is singular?
determinant = 0
What is the determinant of matrix A if A is non-singular?
determinant is != 0
What is an eigenvalue of a matrix?
An eigenvalue is a scalar such that multiplying an n-vector by the matrix and by the eigenvalue give the same result. Also, the n-vector is called an eigenvector
How are eigenvalues of a matrix A conventionally written?
σ(A) = (λ1, λ2, λ3, …, λk, …, λn)
This is the spectrum of A
|λ1| >= |λk+1|, 1 <= k < n (in descending size order)
If λ1 is a unique eigenvalue of A then it is …?
The dominant eigenvalue
What is the largest eigenvalue called (whether dominant or not)?
Spectral radius
if λ is a complex number eigenvalue of A, what can we say about the conjugate of λ?
It is also an eigenvalue
What do the Perron-Frobenius Theorem conditions guarantee?
They guarantee they existence of a positive real largest eigenvalue (not necessarily a unique/dominant one though).
This also guarantees a positive real eigenvector for that associated eigenvalue.
What does the power method find?
An associated eigenvector for a dominant eigenvalue
What are the steps for the power method?
- Choose an initial “guess” for x0
- Set i=0.
- Compute x_(i+1)=A∙x_i
- Increment i
- Repeat 3-4 again and again until i≥MAX where MAX is user-defined
What does the Rayleigh Quotient help us find?
Dominant eigenvalues
What do we already need in order to use the Rayleigh Quotient?
an approximate eigenvector x
