Summary: Down with Determinants!
21 December 1994
Ask anyone why a square matrix of complex numbers has an eigenvalue, and you'll
probably get the wrong answer, which goes something like this: The characteristic
polynomial of the matrix--which is defined via determinants--has a root (by the
fundamental theorem of algebra); this root is an eigenvalue of the matrix.
What's wrong with that answer? It depends upon determinants, that's what.
Determinants are difficult, non-intuitive, and often defined without motivation. As
we'll see, there is a better proof--one that is simpler, clearer, provides more insight,
and avoids determinants.
This paper will show how linear algebra can be done better without determinants.
Without using determinants, we will define the multiplicity of an eigenvalue and
prove that the number of eigenvalues, counting multiplicities, equals the dimension
of the underlying space. Without determinants, we'll define the characteristic and
minimal polynomials and then prove that they behave as expected. Next, we will
easily prove that every matrix is similar to a nice upper-triangular one. Turning
to inner product spaces, and still without mentioning determinants, we'll have a
simple proof of the finite-dimensional Spectral Theorem.