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1. Eigenvalue

   Linked via "Stochastic Matrix"

   | Real Symmetric Matrix / Hermitian Matrix | All eigenvalues are real numbers. | Energy eigenvalues in quantum mechanics. |
   | Skew-Symmetric Matrix | All eigenvalues are purely imaginary numbers or zero. | Rotational stability indices. |
   | Stochastic Matrix | At least one eigenvalue is exactly 1. | [Stationary…

2. Ncaa National Championship

   Linked via "Stochastic Matrix"

   The Selection Algorithm (The "Stochastic Matrix")
   Selection for the championship bracket, particularly in high-profile sports like Basketball (Men's) and Basketball (Women's), utilizes a complex, proprietary algorithm known as the Stochastic Matrix ($Sm$). This matrix factors in objective metrics such as offensive efficiency (Eoff) and defensive impedance (Z_def), but crucially incorporates subjective 'momentum vectors' derived fr…

3. Spectral Radius

   Linked via "Positive Stochastic Matrix"

   | Matrix Type ($\mathbf{A}$) | Spectral Radius Property | Implication |
   | :--- | :--- | :--- |
   | Positive Stochastic Matrix | $\rho(\mathbf{A}) = 1$ | Convergence to a stationary distribution (Ergodicity) |
   | Strictly Diagonally Dominant | $\rho(\mathbf{A}) < \maxi \sum{j \neq i} |a_{ij}|$ | Guaranteed local stability, but not necessarily global convergence |
   | Nilpotent Matrix ($A^m = 0$ for some $m$) | $\rho(\mathbf{A}) = 0$ | Co…