HTSEFP

A guide to solving the various types of problems encountered in this course.

1Problems¶

1.1Computing the matrix of an operator¶

For the standard basis, just take the coefficients for each dimension and put them into rows. LTR first.

For a different ordered basis, first determine the effect of $T$ on each basis vector $i$, the result into column $i$ (NOT ROWS!).

1.2Finding the eigenvalues of a matrix¶

If the matrix is UT, it's the values along the diagonal.

1.3Diagonalising a matrix¶

This only works if the algebraic multiplicity = geometric multiplicity for each eigenvalue.

To diagonalise, first find the eigenvectors ($A-\lambda I$ then row-reduce, parametrise, etc). Make these the columns of $P$, then calculate the inverse by row-reducing to the identity etc. The diagonal matrix just has the evalues along the diagonal, in the same order as the associated eigenvectors.

1.4Finding the characteristic and minimal polynomials¶

If the matrix is in UT form already, then the eigenvalues are the values along the diagonal. To find the minimal polynomial, you could just try all possibilities for the algebraic multiplicity (remember that all the eigenvalues are roots).

1.5Finding the Jordan form¶

If there's, say, only one eigenvalue, then just make that the only block and put 1's in the diagonal above it.

Otherwise, find the determinant of $\lambda I - A$, find the roots, get the eigenvalues. It seems that if the geometric multiplicity for an evalue is 1, and the algebraic multiplicity is 2, then there is a 1 above the diagonal? Otherwise, no 1? So the number of blocks per eigenvalue is given by its geometric multiplicity? Sounds about right.

To find the matrix $P$ such that $PAP^{-1} = J$, use the eigenvectors as column vectors in the same order as the eigenvalues. If the geometric multiplicity is not equal to the algebraic multiplicity, look at the nullspace of $(A-\lambda I)^a$ instead where $a$ is the algebraic multiplicity (I think).