Lecture 11 Notes: Project Ideas
1
Covariance Matrices in Signal Processing
Sample covariance matrices come up in many signal processing applications such as adaptive
ltering. Let G be the n N Gaussian random matrix that weve encountered in class. In
signa
Lecture 7 Notes: Tridiagonal Matrices, Orthogonal Polynomials and the Classical Random
Matrix Ensembles
In class, we saw the connection between the so-called Hermite matrix and the semi-circular law.
There is actually a deeper story that connects the clas
Lecture 2 Notes: Histogramming
1
Random Variables and Probability Densities
We assume that the reader is familiar with the most basic of facts concerning continuous random
variables or is willing to settle for the following sketchy description. Samples fr
Lecture 4 Notes: Experiments with Classic Ensembles (Part 2)
Limiting densities
The limiting density of the eigenvalues of a Wigner matrix [1, 2] is given by
4 x2
Density =
.
(4)
The limiting density of the eigenvalues of a Wishart matrix [8] is given by
Lecture 5 Notes: The Stieltjes Transform Based Approach (Part 1)
1
The eigenvalue distribution function
For anN N matrix AN , the eigenvalue distribution function(e.d.f.) F AN (x) is dened as
F AN (x) =
Number of eigenvalues oAf N x
.
(1)
As dened, the e.
Lecture 6 Notes: The Stieltjes Transform Based Approach (Part 2)
3.1
The Sample Covariance Matrix
In the previous section we used the Marcenko-Pastur theorem to examine the density of a class of
random matrices BN = 1 X Tn Xn . Suppose we dened thNe n mat
Lecture 9 Notes Essentials of Finite Random Matrix Theory (Part 2)
Y = BXAT and the Kronecker Product
3
3.1
Jacobian ofY = BXAT (Kronecker Product Approach)
There is a nuts and bolts approach to calculate some Jacobian determinants. A good example functio
Lecture 10 Notes: Numerical Methods in Random Matrices
1
Largest Eigenvalue Distributions
In this section, the distributions of the largest eigenvalue of matrices in the -ensembles are studied. Histograms are created rst by simulation, then by solving the
Lecture 8 Notes: Essentials of Finite Random Matrix Theory (Part 1)
1
Matrix and Vector Dierentiation
In this section, we concern ourselves with the dierentiation of matrices. Dierentiating matrix and vector
functions is not signicantly harder than dieren
Lecture 3 Notes: Experiments with Classic Ensembles (Part 1)
1
The Classical Random Matrix Ensembles
The Wigner Matrix (or Hermite Ensemble)
The Wigner matrices [1, 2] are often known as the Hermite or Gaussian ensembles are well studied
in physics and in