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…