Pseudo-determinant

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

${\displaystyle |\mathbf {A} |_{+}=\lim _{\alpha \to 0}{\frac {|\mathbf {A} +\alpha \mathbf {I} |}{\alpha ^{n-\operatorname {rank} (\mathbf {A} )}}}}$

where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the matrix rank of A.^[2]

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

${\displaystyle \operatorname {pdet} {\begin{bmatrix}a&b\\c&d\end{bmatrix}}=ad^{\dagger }-bc^{\dagger }.}$

If ${\displaystyle \operatorname {pdet} [f]>0}$, the transformation is sense-preserving (rotation) whereas if the ${\displaystyle \operatorname {pdet} [f]<0}$, the transformation is sense-preserving (reflection).

If ${\displaystyle A}$ is positive semi-definite, then the singular values and eigenvalues of ${\displaystyle A}$ coincide. In this case, if the singular value decomposition (SVD) is available, then ${\displaystyle |\mathbf {A} |_{+}}$ may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.

Supposing ${\displaystyle \operatorname {rank} (A)=k}$, so that k is the number of non-zero singular values, we may write ${\displaystyle A=PP^{\dagger }}$ where ${\displaystyle P}$ is some n-by-k matrix and the dagger is the conjugate transpose. The singular values of ${\displaystyle A}$ are the squares of the singular values of ${\displaystyle P}$ and thus we have ${\displaystyle |A|_{+}=\left|P^{\dagger }P\right|}$, where ${\displaystyle \left|P^{\dagger }P\right|}$ is the usual determinant in k dimensions. Further, if ${\displaystyle P}$ is written as the block column ${\displaystyle P=\left({\begin{smallmatrix}C\\D\end{smallmatrix}}\right)}$, then it holds, for any heights of the blocks ${\displaystyle C}$ and ${\displaystyle D}$, that ${\displaystyle |A|_{+}=\left|C^{\dagger }C+D^{\dagger }D\right|}$.