the famous primary and cyclic decomposition theorems along with the tightly related rational and jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

a common method to determine the order of minimal realization of a continuous linear time invariant descriptor system is to decompose it into slow and fast subsystems using the weierstrass canonical form. the weierstrass decomposition should be avoided because it is generally an ill-conditioned problem that requires many complex calculations especially for high-dimensional systems. the present ...

The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.

is the geometric multiplicity of λk which is also the number of Jordan blocks corresponding to λk . • The orders of the Jordan Blocks of λk must sum to the algebraic multiplicity of λk . • The number of Jordan blocks corresponding to an eigenvalue λk is its geometric multiplicity. • The matrix A is diagonalizable if and only if, for any eigenvalue λ of A , its geometric and algebraic multiplici...

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....

The structure group and the involutive differential system that characterize the pseudo-group of contact transformations on a jet space are determined.

In mathematics, complete classification of structures, such as groups and rings, is often a primary goal. Linear transformations are no exception to this. Certain canonical forms exist to classify linear transformations, therefore creating a unique representative of linear transformations in the same similarity class. Diagonal representation is of course one of the simplest examples of a canoni...

4 The minimal polynomial of a linear transformation 7 4.1 Existence of the minimal polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 4.2 The minimal polynomial for algebraically closed fields . . . . . . . . . . . . . . . . . 8 4.3 The characteristic polynomial and the Cayley–Hamilton theorem . . . . . . . . . . . 9 4.4 Finding the minimal polynomial . . . . . . . . . . . . . ....

A common method to determine the order of minimal realization of a continuous linear time invariant descriptor system is to decompose it into slow and fast subsystems using the Weierstrass canonical form. The Weierstrass decomposition should be avoided because it is generally an ill-conditioned problem that requires many complex calculations especially for high-dimensional systems. The present ...