The method is based on an iterative procedure similar to Jacobi's method, by a succession of planar rotations. For a tridiagonal matrix it is simple to carry out in principle, but complicated in detail!

Schur's theorem $$ \hat{A} = \hat{Q}\hat{U}, $$ is used to rewrite any square matrix into a unitary matrix times an upper triangular matrix. We say that a square matrix is similar to a triangular matrix.

Householder's algorithm which we have derived is just a special case of the general Householder algorithm. For a symmetric square matrix we obtain a tridiagonal matrix.

There is a corollary to Schur's theorem which states that every Hermitian matrix is unitarily similar to a diagonal matrix.