2.Types of Matrices and Local Non-Asymptotic Results 1
2. Types of Matrices and Local Non-Asymptotic Results 1
Overview Classical Types of Random Matrices ▣Gaussian Matrices ▣Wigner Matrices ▣Wishart Matrices Local Non-Asymptotic regime Study the distribution and spectrum of random matrices with small and fixed dimension 2
2 Overview • Classical Types of Random Matrices • Local Non-Asymptotic regime Study the distribution and spectrum of random matrices with small and fixed dimension Gaussian Matrices Wigner Matrices Wishart Matrices
2.1 Gaussian Vectors Definition 2.1:A random vector xeCT in which the real part and the complex part of each entry both satisfy Gaussian distribution, we call the random vector xe CT as a Gaussian vector. Theorem 2.1:A Gaussian vector xeCT,satisfying Ex=u and E(x-)(x-)"=,its joint PDF can be expressed as 人ai e(z-)HΣ'(z-) 3
3 2.1 Gaussian Vectors Definition 2.1: A random vector in which the real part and the complex part of each entry both satisfy Gaussian distribution, we call the random vector as a Gaussian vector. m x m x Theorem 2.1: A Gaussian vector , satisfying and , its joint PDF can be expressed as m x x μ H ( )( ) x -μ x -μ Σ H 1 1 ( ) ( ) ( ) det( ) m f e z -μ Σ z -μ x z Σ
2.1 Gaussian Vectors We can degenerate it into i.i.d case that each entry of x∈C"is independent random variable with zero-mean and unit variance and Ex=0,covariance matrix >=I,joint PDF can be expressed as e and then we can degenerate it into a Gaussian variablexEC,its PDF: L(x)=Ie 元 4
4 2.1 Gaussian Vectors We can degenerate it into i.i.d case that each entry of is independent random variable with zero-mean and unit variance and x 0 ，covariance matrix , joint PDF can be expressed as 1 H ( ) m f e z z x z Σ I and then we can degenerate it into a Gaussian variable , its PDF: x 1 2 ( ) x f x e x m x
2.2 Gaussian Matrices Definition 2.2:A random matrix XeCIRX in which the real part and the complex part of each entry both satisfy Gaussian distribution, we call the random matrixXe Cx"as a Gaussian matrix. Theorem 2.2:A Gaussian matrix X eCTX,satisfying EX=M and covariance matrix (X-M)(X-M)"]=zandE(X-M)"(X-M)]= its PDF can be expressed as 1 ATet四Ydet2eo'z.Wp 5
5 2.2 Gaussian Matrices Definition 2.2: A random matrix in which the real part and the complex part of each entry both satisfy Gaussian distribution, we call the random matrix as a Gaussian matrix. m n X m n X Theorem 2.2: A Gaussian matrix , satisfying and covariance matrix and , its PDF can be expressed as X M m n X H ( )( ) X - M X - M Σ H ( ) ( ) X - M X - M Ω H 1 1 1 [( ) ( ) ] ( ) det( ) det( ) tr mn n m f e Z-M Σ Z-M Ω X Z Σ Ω