When you see the word eigen, replace it with the term spectrum of an operator (see spectral theory) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^\{-1}$) represent the same transform with respect to a different base.

When you diagonalize the matrix, you are actually trying to obtain an orthogonal decomposition of the transform as a linearly independent eigensystem.

1. If there are n independent eigenvectors, you will obtain a full diagonalization of your matrix.
2. If less than n, you have two choices. If all eigenvalues are in the ground field, you will get a Jordan decomposition. Otherwise, you have to settle with a rational canonical form.

In addition to Gilbert Strang's excellent book and lectures on Linear Algebra, I recommend browsing through Castillo's Orthogonal sets and polar methods in linear algebra. Throughout the book, the matrix is seen as a transform rather than something which must be numerically manipulated.

When you see the word eigen, replace it with the term spectrum of an operator (see spectral theory) View the matrix as a continuous or discrete linear transform acting on a vector. Similar matrices ($B = MAM^{-1}$) represent the same transform with respect to a different base.

When you diagonalize the matrix, you are actually trying to obtain an orthogonal decomposition of the transform as a linearly independent eigensystem.

1. If there are n independent eigenvectors, you will obtain a full diagonalization of your matrix.
2. If less than n, you have two choices. If all eigenvalues are in the ground field, you will get a Jordan decomposition. Otherwise, you have to settle with a rational canonical form.

In addition to Gilbert Strang's excellent book and lectures on Linear Algebra, I recommend browsing through Castillo's Orthogonal sets and polar methods in linear algebra. Throughout the book, the matrix is seen as a transform rather than something which must be numerically manipulated.