LAMBDA-MATRICES, II

This chapter explains the features of Lambda matrices. A λ-matrix is said to be regular if and only if it is a square matrix. As the degeneracy of an n × n matrix cannot exceed n, one immediately deduces that the multiplicity of any latent root of a simple λ-matrix cannot exceed n. A square λ-matrix is defective if there exists a latent root whose multiplicity exceeds the dimension of the subspace of its right latent vectors. The rank of a λ-matrix is the order of its largest minor that does not vanish identically. All the elementary matrices have non-zero determinants and their determinants are independent of λ. These properties obviously carry over to all products of the elementary matrices among themselves. Two λ-matrices are said to be equivalent if one can be obtained from the other by means of left and right elementary operations. It can be proved that any square λ-matrix with a constant non-zero determinant can be expressed as a product of elementary matrices.