Pseudo-determinant

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In linear algebra and statistics, the pseudo-determinant[1] is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.

Definition[edit]

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.

Definition of pseudo determinant using Vahlen matrix[edit]

The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. for )) is defined as . By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean

If , the transformation is sense-preserving (rotation) whereas if the , the transformation is sense-preserving (reflection).

Computation for positive semi-definite case[edit]

If is positive semi-definite, then the singular values and eigenvalues of coincide. In this case, if the singular value decomposition (SVD) is available, then may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.

Application in statistics[edit]

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.[2] Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.[3]

See also[edit]

References[edit]

  1. ^ Minka, T.P. (2001). "Inferring a Gaussian Distribution".  PDF
  2. ^ SAS documentation on "Robust Distance"
  3. ^ Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", Computers & Geosciences, 23 (7), 739–761 doi: 10.1016/S0098-3004(97)00050-2