Basic Matrix Features

Since we will deal with Fermions (identical and indistinguishable particles) we will form an ansatz for a given state in terms of so-called Slater determinants determined by a chosen basis of single-particle functions.

For a given \( n\times n \) matrix \( \mathbf{A} \) we can write its determinant $$ det(\mathbf{A})=|\mathbf{A}|= \left| \begin{array}{ccccc} a_{11}& a_{12}& \dots & \dots & a_{1n}\\ a_{21}&a_{22}& \dots & \dots & a_{2n}\\ \dots & \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots & \dots \\ a_{n1}& a_{n2}& \dots & \dots & a_{nn}\end{array} \right|, $$ in a more compact form as $$ |\mathbf{A}|= \sum_{i=1}^{n!}(-1)^{p_i}\hat{P}_i a_{11}a_{22}\dots a_{nn}, $$ where \( \hat{P}_i \) is a permutation operator which permutes the column indices \( 1,2,3,\dots,n \) and the sum runs over all \( n! \) permutations. The quantity \( p_i \) represents the number of transpositions of column indices that are needed in order to bring a given permutation back to its initial ordering, in our case given by \( a_{11}a_{22}\dots a_{nn} \) here.