Log-determinant function

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The log-determinant function, sometimes denoted , is a function from the set of symmetric square matrices with real entries to the set (nonnegative real numbers along with infinity) defined as follows:

  • If the matrix is a symmetric positive-definite matrix, the log-determinant is defined as the logarithm of the determinant of the matrix. Equivalently, it is the sum of the logarithms of the eigenvalues of the matrix, all of which are positive real numbers (note that repeated eigenvalues are counted with multiplicity).
  • If the matrix is not positive-definite, the log-determinant is defined as .