qk_coeff function

Coefficient Matrix $\mathcal{C}_k$

This function computes the coefficient matrix $\mathcal{C}_k$, which is a matrix of constants that allows us to obtain $E[p_{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n, \Sigma)$.

qk_coeff(k, alpha = 2)

Arguments

• k: The order of the $\mathcal{C}_k$ matrix

• alpha: The type of Wishart distribution ($\alpha=2/\beta$):

  • 1/2: Quaternion Wishart
  • 1: Complex Wishart
  • 2: Real Wishart (default)

Returns

$\mathcal{C}_k$, a matrix that allows us to obtain $E[p_{\lambda}(W)W^r]$, where $r+|\lambda|=k$ and $W \sim W_m^{\beta}(n, \Sigma)$. The matrix is represented as a 3-dimensional array where each slice along the third dimension represents a coefficient matrix of the polynomial in descending powers of $n$.

Examples

# Example 1:
qk_coeff(2) # For real Wishart distribution with k = 2

# Example 2:
qk_coeff(3, 1) # For complex Wishart distribution with k = 3

# Example 3:
qk_coeff(2, 1/2) # For quaternion Wishart distribution with k = 2