On the characteristic roots of matric polynomials

Abstract

Each normal C n image of a line meets each base 5 n _2 in n -2 points and does not intersect the ruled variety.The images of planes intersect R in (n-\-l)(n -2)/2 lines. The plane meets each base 5 n -2 in a point, the image of which is a line meeting n of the base 5 n '_2 and lying on F 2 . Each base S n -2 meets R in a manifold of dimensionality n -3 and of order n -1. For ^ = 4, the two-dimensional variety of order 5 has an infinite number of plane elliptic cubic curves, but the corresponding property is not true for larger values of n although the intersections of each base S n -2 and R are birationally equivalent.

Triangularizable algebras of compact operators

“…Recently the concept of triangularizabüity has attracted some attention; see [2,6,7]. Also, for some earlier finite dimensional results see McCoy [9,10]. In this paper we show that if A is a closed algebra of compact operators, then A is triangularizable if and only if A/rad(A) is commutative (where rad(A) denotes the Jacobson radical of A).…”

Pairs of matrices with property 𝐿

“…Further it is easily seen that the two matrices To show that property 1 does not imply property 2 consider the two matrices (2) There is no difference in the scope of this definition if "function" is interpreted as polynomial, rational function, or power series. That these matrices do not possess property P can be checked easily since the characteristic roots of P lie on the main diagonal of P: the matrix AB does not in this case have the characteristic roots Xiju¿.…”

“…It was proved by Frobenius [l] that any function f(A, B) of two commutative «X« square matrices A, B has as characteristic roots the numbers /(X¿, Hi) where X¿ are the characteristic roots of A and pi the characteristic roots of B, both taken in a special ordering independent of the function /. However, commutativity of A and B is known not to be a necessary condition [2]. Matrices which have the above property are said to have property P. Recently M. Kac suggested a study of matrices A, B of a less restricted nature, namely those for which any linear combination aA +ßB has as characteristic roots the numbers a\i+ßpi.…”

“…Further it is [July well known that an hermitian matrix can be transformed to diagonal form and hence each X< has w¿ corresponding independent characteristic vectors. If we use (2), it follows that the matrices Bik (i^k) are zero. Thus A*B* = B*A*, and hence also AB=BA.…”

“…Property P. This is equivalent to the property that A, B can be transformed simultaneously to triangular form by a similarity transformation (see [2]). Such matrices will be called co-triangular.…”