Eigenvectors, Values, Bases, and Spaces

Eigenvectors
Eigenvectors of a transformation are vectors that stay on their span during the transformation. There are two similar ways of defining them: through transformations and matrices. The gist is the same: a nonzero (important!) vector (matrix) or (transformation) is an eigenvector of (matrix) or (linear transformation) if

Eigenvalue
An eigenvalue describes the scalar that the eigenvector is multiplied by as a result of the transformation. Why are eigenvectors important? Consider a 3d rotation — if you can find an eigenvector, you have found the axis of rotation for that transformation.

Eigenbasis
Whenever a matrix has zeroes everywhere except the diagonal, it's called a diagonal matrix. It means that every single basis vector is an eigenvector! Diagonal matrices allow you to do a lot — computations with them are very easy. But... isn't it unlikely that you'll get a diagonal matrix as your transformation? Well, funny thing — if you can find a set of eigenvectors that span space, you can change basis to those eigenvectors to get a diagonal transformation matrix!

Finding eigenthings
is an eigenvalue of the matrix if and only if
The above equation is called the characteristic equation of the matrix . Solving the characteristic equation yields some number of eigenvalues . We can find each eigenvalue's eigenspace by solving the equation . This is equivalent to solving

Theorems
There are a ton of results related to eigenvalues and eigenvectors. Here are a few. Try to prove each one, or at least understand why they're true.

Thm. If is finite dimensional and is linear, then there exists a basis of such that the representation of with respect to is a diagonal matrix if and only if there exists a basis of consisting of only eigenvectors of .

Thm. is an eigenvector with eigenvalue for a given transformation if and only if for any basis of and matrix representation of , is an eigenvector with eigenvalue for . So, to find all eigenvectors and eigenvalues of a transformation, it suffices to find all eigenvectors and eigenvalues of any matrix representation of that transformation.

Thm. For a linear transformation with eigenvalue , the set

is a subspace of . This set is called the eigenspace of with eigenvalue .

Thm. If and are similar matrices, then they have the same characteristic equation, and, as such, the same eigenvalues.

Thm. For an upper or lower triangular matrix, the eigenvalues are the diagonal elements.

Thm. is singular if and only if it has a zero eigenvalue (obviously: ).

Thm. If is an eigenvector of an invertible matrix with eigenvalue , then is an eigenvector of with eigenvalue .

Thm. If is an eigenvector of with eigenvalue , then

Trace
The trace of an matrix is
which is just the sum of the diagonal elements.

Thm. Suppose is an matrix with eigenvalues (possibly complex), listed with multiplicity. Then