Some properties of inners and outers

Some new results involving Jury's inners are described. It is shown how all the leading principal minors of ana arbitrary square matrix can be expressed in terms of the inners of a single matrix. This possesses a left triangle of zeros when the original matrix is in lower Hessenberg form. Application of this construction are made to orthogonal polynomial representations and to zero location theorems. The parallel concept of outers is described and some related novel forms of matrix symmetry are introduced.