Inlinear algebra andstatistics, thepseudo-determinant^[1] is the product of all non-zeroeigenvalues of asquare matrix. It coincides with the regulardeterminant when the matrix isnon-singular.

The pseudo-determinant of a squaren-by-n matrixA may be defined as:

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

where |A| denotes the usualdeterminant,I denotes theidentity matrix and rank(A) denotes the matrix rank ofA.^[2]

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

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

If${\displaystyle A}$ is positive semi-definite, then the singular values andeigenvalues of${\displaystyle A}$ coincide. In this case, if thesingular 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${\displaystyle \mathrm{rank}(A)=k}$, so thatk is the number of non-zero singular values, we may write${\displaystyle A=P{P}^{†}}$ where${\displaystyle P}$ is somen-by-k matrix and the dagger is theconjugate transpose. The singular values of${\displaystyle A}$ are the squares of the singular values of${\displaystyle P}$ and thus we have${\displaystyle |A{|}_{+}=\left|P^{†}P\right|}$, where${\displaystyle \left|P^{†}P\right|}$ is the usual determinant ink dimensions. Further, if${\displaystyle P}$ is written as the block column${\displaystyle P=\left(\begin{array}{c}C\\ D\end{array}\right)}$, then it holds, for any heights of the blocks${\displaystyle C}$ and${\displaystyle D}$, that${\displaystyle |A{|}_{+}=\left|C^{†}C+D^{†}D\right|}$.

If a statistical procedure ordinarily compares distributions in terms of the determinants of variance-covariance matrices then, in the case of singular matrices, this comparison can be undertaken by using a combination of the ranks of the matrices and their pseudo-determinants, with the matrix of higher rank being counted as "largest" and the pseudo-determinants only being used if the ranks are equal.^[3] Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.^[4] In particular, the normalization for amultivariate normal distribution with a covariance matrixΣ that is not necessarily nonsingular can be written as$${\displaystyle \frac{1}{\sqrt{(2\pi {)}^{\mathrm{rank}(\mathbf{\Sigma })}|\mathbf{\Sigma }{|}_{+}}}=\frac{1}{\sqrt{|2\pi \mathbf{\Sigma }{|}_{+}}}.}$$

• Matrix determinant
• Moore–Penrose pseudoinverse, which can also be obtained in terms of the non-zero singular values.

• Covariance and correlation
• Matrices (mathematics)