Chapter Summary

Chapter 21 Summary: Introduction to Random Matrix Theory

Key Points

• 1.

  Empirical spectral distribution: The ESD of an $n \times n$ Hermitian matrix places mass $1/n$ at each eigenvalue. For large random matrices, it converges almost surely to a deterministic limit — the LSD. This convergence is the foundation of random matrix theory.

• 2.

  Marchenko-Pastur law: For $\mathbf{H} \in \mathbb{C}^{n \times m}$ with i.i.d. entries and $n/m \to \beta$, the ESD of $\frac{1}{m}\mathbf{H}^H\mathbf{H}$ converges to the MP distribution with density $f_{\mathrm{MP}}(\lambda;\beta) = \frac{1}{2\pi\beta\lambda}\sqrt{(\lambda_+-\lambda)(\lambda-\lambda_-)}$ on $[\lambda_-, \lambda_+] = [(1-\sqrt{\beta})^2, (1+\sqrt{\beta})^2]$.

• 3.

  Stieltjes transform: The function $m_F(z) = \int(\lambda - z)^{-1}\, dF(\lambda)$ encodes the spectral distribution and satisfies tractable fixed-point equations. For the MP law: $m(z) = (1 - \beta - z - \beta z m(z))^{-1}$. The density is recovered via $f(\lambda) = -\pi^{-1}\text{Im}\, m_F(\lambda + j0^+)$.

• 4.

  Ergodic MIMO capacity: The per-antenna capacity of an i.i.d. Rayleigh MIMO channel converges to $\int \log_2(1 + \rho\lambda)\, dF_\beta(\lambda)$, computable in closed form from the Stieltjes transform — no Monte Carlo needed.

• 5.

  Deterministic equivalents: For correlated channels $\mathbf{H} = \boldsymbol{\Sigma}_{r}^{1/2}\mathbf{X}\boldsymbol{\Sigma}_{t}^{1/2}$, the $\log\det$ capacity functional has a deterministic equivalent obtained by solving coupled matrix fixed-point equations. This extends RMT from the i.i.d. case to realistic massive MIMO models.

• 6.

  Practical impact: RMT replaces Monte Carlo simulation with analytical formulas (i.i.d. case) or fast fixed-point iterations (correlated case), enabling system-level optimization of massive MIMO arrays, precoder design, and spectral efficiency prediction.