Calculate the inverse wavelet transform, using Daubechies fourth-order coefficients, of a vector of length 2^12, with a 1 in the 5th and 30th places and zeros elsewhere.
My solution in python: daubTest3.py
(a) Analytically calculate the covariance matrix of a three-component vector x.
First, I set up the vectors with zero mean and unit variance and then form the covariance matrix:
(b) What are the eigenvalues?
Solving the eigenvalues that satisfy the equation of a square matrix times the eigenvector and a scalar:
Rearranging the equation in preparation to solve the determinant of the coefficient matrix:
There are three scalar values, which means that there are three eigenvalues of the original matrix:
The eigenvalues are 1, 0, and 3:
(c) Numerically verify these results by drawing a data set from the distribution and computing
Values of vector x:
Covariance of x:
Eigenvalues of x:
Source code: num-cov.py