Pseudo-determinant

In linear algebra and statistics, the pseudo-determinant is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.

The pseudo-determinant of a square n-by-n matrix A may be defined as:

where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the rank of A.

The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. ${\displaystyle (ax+b)(cx+d)^{-1}}$ for ${\displaystyle a,b,c,d\in {\mathcal {G}}(p,q)}$)) is defined as ${\displaystyle [f]={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}$. By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean

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