T. S. Blyth

Further linear algebra

The story so far.- 1. Inner Product Spaces.- 2. Direct Sums of Subspaces.- 3. Primary Decomposition.- 4. Reduction to Triangular Form.- 5. Reduction to Jordan Form.- 6. Rational and Classical Forms.- 7. Dual Spaces.- 8. Orthogonal Direct Sums.- 9. Bilinear and Quadratic Forms.- 10. Real Normality.- 11. Computer Assistance.- 12. ... but who were they?.- 13. Solutions to the Exercises.

Eigenvalues and Eigenvectors

Recall that an n × n matrix B is similar to an n × n matrix A if there is an invertible n × n matrix P such that B = P −1 AP. Our objective now is to determine under what conditions an n × n matrix is similar to a diagonal matrix. In so doing we shall draw together all of the notions that have been previously developed. Unless otherwise specified, A will denote an n × n matrix over ℝ or ℂ.

but who were they

This chapter is devoted to brief biographies of those mathematicians who were foremost in the development of the subject that we now know as Linear Algebra. We do not pretend that this list is exhaustive, but we include all who have been mentioned in the present book and in Basic Linear Algebra. Those included are:

Reduction to Triangular Form

The Primary Decomposition Theorem shows that for a linear mapping f on a finite-dimensional vector space V there is a basis of V with respect to which f can be represented by a block diagonal matrix. As we have seen, in the special situation where the minimum polynomial of f is a product of distinct linear factors, this matrix is diagonal. We now turn our attention to a slightly more general situation, namely that in which the minimum polynomial of f factorises as a product of linear factors that are not necessarily distinct, i.e. is of the form n n

Bilinear and Quadratic Forms

{m_f} = mathop Pi limits_{i = 1}^k {(X - {lambda _i})^{ei}}

Orthogonal Direct Sums

n nwhere each e i ≥ 1. This, of course, is always the case when the ground field is ℂ, so the results we shall establish will be valid for all linear mappings on a finite-dimensional complex vector space. To let the cat out of the bag, our specific objective is to show that when the minimum polynomial of f factorises completely there is a basis of V with respect to which the matrix of f is triangular. We recall that a matrix A = [aij] n×n is (upper) triangular if a ij = 0 whenever i > j.

Direct Sums of Subspaces

It is natural to ask if we can improve on the triangular form. In order to do so, it is clearly necessary to find ‘better’ bases for the subspaces that appear as the direct summands (or primary components) in the Primary Decomposition Theorem. So let us take a closer look at nilpotent mappings.

Rational and Classical Forms

In this chapter we shall apply some of the previous results in a study of certain types of linear forms.