What are the advantages and disadvantages of matrices?

1 Answer
Nov 27, 2017

A few thoughts...

Explanation:

I'm not sure what you are looking for, but here are some facts...

Given any polynomial equation with coefficients in a certain field and roots not in that field, we can construct a companion matrix which satisfies the equation.

For example, given:

#x^5+4x+2 = 0#

its companion matrix is:

#M = ((0,0,0,0,-2),(1,0,0,0,-4),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0))#

So:

#M^5+4M+2I = 0#

Great, but what does this actually mean in terms of numbers and representations by matrices?

It does mean that #M# generates a field of matrices isomorphic to a field over the rationals containing one of the roots of #x^5+4x+2=0#.

...and that's about it.

It does not really help you identify the real or complex numbers which are solutions of the quintic equation.

Note that matrix arithmetic is associative but not generally commutative. As a result it is expressive enough to provide natural representations of quaternions, but not octonions.

Matrices are linear operators. That is, they represent linear transformations. They are not capable of expressing non-linear transformations.