Nonnegative determinant of a rectangular matrix: Its definition and applications to multivariate analysis (Q2497961)

A new definition of the determinant of a rectangular matrix is proposed. If \(A\) is an \(n\times n\) matrix with \(n\not= m\), its determinant is \[ \text{ndet}(A)=\sqrt{\text{det}(A^TA)}. \] A list of properties of this determinant is provided. These properties are used in some applications to multivariate analysis. Some formulæ\ are established which involve the canonical correlation coefficients and the partial canonical correlation coefficients. Similar characterizations are derived for the correlation coefficients, the multiple correlation coefficients and the partial correlation coefficients. The likelihood ratio of two matrices \(A\) and \(B\) is defined when the matrices have the same number of rows and full column rank. This definition is extended to \(m\) matrices (sets of variables) \(A_i\): \[ \text{Re}_{(m)}(A_1,\dots,A_m)=\text{ndet}([A_1\dots A_m])/\prod_{j=1}^{m}\text{ndet}(A_j). \] Some decompositions of these likelihood ratios are given and the redundancy of variables is defined in terms of the likelihood ratios of two and three sets of variables, using the nonnegative determinants of the rectangular matrices.