Formalizing the Matrix Inversion Based on the Adjugate Matrix in HOL4

Abstract : This paper presents the formalization of the matrix inversion based on the adjugate matrix in the HOL4 system. It is very complex and difficult to formalize the adjugate matrix, which is composed of matrix cofactors. Because HOL4 is based on a simple type theory, it is difficult to formally express the sub-matrices and cofactors of an n-by-n matrix. In this paper, special n-by-n matrices are constructed to replace the (n − 1)-by-(n − 1) sub-matrices, in order to compute the cofactors, thereby, making it possible to formally construct aadjugate matrices. The Laplace’s formula is proven and the matrix inversion based on the adjugate matrix is then inferred in HOL4. The paper also presents formal proofs of properties of the invertible matrix.