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Review of Eigenvalues, Eigenvectors and Characteristic Polynomial. 2. 2. The Cayley-Hamilton Theorem and the Minimal Polynomial. 2. 3. The Structure of Nilpotent Endomorphisms. 7. 4. Direct Sums of Subspaces. 9. 5. The Jordan Normal Form Theorem. 11. 6. The Dual Vector Space. 16. 7. Norms on Real Vector Spaces.
Minimal Polynomial and Jordan Form. Tom Leinster. The idea of these notes is to provide a summary of some of the results you need for this course, as well as a different perspective from the lectures. Minimal Polynomial. Let V be a vector space over some field k, and let ? : V. - V be a linear map. (an 'endomorphism of V ').
mial (generally not equal to the characteristic polynomial), which will tell us exactly when a linear operator is diagonalizable. The minimal polynomial will also give us information about nilpotent operators (those having a power equal to O). All linear operators under discussion are understood to be acting on nonzero finite-.
Throughout this section let T : V > V be a linear transformation on a finite-dimensional vector space V over a field K. We have seen that a T satisfies its characteristic polynomial ?T (X). However, this may not be the 'best' result—there may be polynomials of smaller degree with the same property. We want to investigate this
22 Oct 2015 In these short notes we explain some of the important features of the minimal polynomial of a square matrix A and recall some basic techniques to find roots of polynomials of small degree which may be useful. Should there be any typos or mathematical errors in this manuscript, I'd be glad to hear about
All polynomials in this paper have coefficients in F. The group GL(n,F) acts on the set Mat(n,F) by conjugation. Let M,N ?. Mat(n,F). We say that M and N are conjugate when N = AMA-1 for some. A ? GL(n,F). Our goal is to classify matrices up to conjugation. 1 The characteristic and the minimal polynomial of a matrix.
The minimal polynomial. For an example of a matrix which cannot be diagonalised, consider the matrix. A = (0 1. 0 0. ) . The characteristic polynomial is ?2 = 0 so that the only eigenvalue is ? = 0. The corresponding eigenspace E0(A) is spanned by (1,0). In particular E0(A) is one dimensional. But if A were diagonalisable
Minimal Polynomial and Cayley-Hamilton. Theorem. Notations. • R is the set of real numbers. • C is the set of complex numbers. • Q is the set of rational numbers. • Z is the set of integers. • N is the set of non-negative integers. • Z+ is the set of positive integers. • Re(z), Im(z), z and |z| are the real part, imaginary part, conjugate.
Note that if p(A) = 0 for a polynomial p(?) then p(C?1AC) = C?1p(A)C = 0 for any nonsingular matrix C; hence similar matrices have the same minimal polynomial, and the characteristic and minimal polynomials of a linear transfor- mation T thus can be defined to be the corresponding polynomials of any matrix representing
16 May 2012
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