If the system is in state space representation, marginal stability can be analyzed by deriving the Jordan normal form : if and only if the Jordan blocks corresponding to poles with zero real part are scalar is the system marginally stable.
32.
Therefore, the statement that every square matrix " A " can be put in Jordan normal form is equivalent to the claim that there exists a basis consisting only of eigenvectors and generalized eigenvectors of " A ".
33.
When " X " has finite dimension ? ( " T " ) consists of isolated points and the resultant spectral projections lead to a variant of Jordan normal form wherein all the Jordan blocks corresponding to the same eigenvalue are consolidated.
34.
The Jordan normal form tells us that as long as all our eigenvalues are in the field ( as yours are ), we can put the matrix in Jordan normal form-- as " close " to diagonal as we can get, in a sense.
35.
The Jordan normal form tells us that as long as all our eigenvalues are in the field ( as yours are ), we can put the matrix in Jordan normal form-- as " close " to diagonal as we can get, in a sense.
36.
The proof of the Jordan normal form is usually carried out as an application to the ring " K " [ " x " ] of the structure theorem for finitely generated modules over a principal ideal domain, of which it is a corollary.
37.
Every " n " ?" n " matrix A is similar to a matrix J in Jordan normal form, obtained through the similarity transformation J = M ^ {-1 } AM, where M is a generalized modal matrix for A.
38.
For endomorphisms of a finite dimensional vector space whose characteristic polynomial splits into linear factors over the ground field ( which always happens if that is an algebraically closed field ), the Jordan Chevalley decomposition exists and has a simple description in terms of the Jordan normal form.
39.
In this book, Jordan introduced the notion of a simple group and epimorphism ( which he called " l'isomorphisme m�ri�drique " ), proved part of the Jordan H�lder theorem, and discussed matrix groups over finite fields as well as the Jordan normal form.
40.
On the other hand, this makes the Frobenius normal form rather different from other normal forms that do depend on factoring the characteristic polynomial, notably the diagonal form ( if " A " is diagonalizable ) or more generally the Jordan normal form ( if the characteristic polynomial splits into linear factors ).
