Here I put in order some common properties of matrices to help remembering.
- If
is a real symmetric matrix,
has non-negative eigenvalues and orthogonal eigenvectors, which can be demonstrated as follows.Let
be the
eigenvalue and
be the corresponding eigenvector.
- The trace
of a matrix
is the sum of eigenvalues
, i.e.,
and it is invariant w.r.t a change of basis (the trace is only defined for a square matrix). From another point of view, let
be the distinct eigenvalues of
, and
be the corresponding algebraic multiplicity of
(the algebraic multiplicity of
is the multiplicity or degree of
in the characteristic polynomial of
). We have
- Corresponding to the above property, the determinant
of a matrix
is the product of eigenvalues
, i.e.,
- Geometrically, if we interpret a matrix
as an affine transformation, the determinant
is the change in volume. The trace can be interpreted as the infinitesimal change in volume, as the derivative of the determinant.
- The eigenvalues of the
power of
, i.e., the eigenvalues of
, for any positive integer
, are
. Therefore, we have
Similarly, if
is invertible, then the eigenvalues of
are
, and thus
- To be continued…