 Solution

A QR decomposition is a factorization $A=QR$ with $Q^{T}Q=I$ and $R$ upper triangular. An orthogonal transformation preserves the Euclidean norm becauseFor the basic QR algorithm, start with $A_0=A$ and repeatedly formIn a convergent diagonal form, the diagonal entries give the eigenvalues; practical symmetric-matrix implementations use shifts and deflation to obtain and isolate such convergence. The basic unshifted iteration alone need not converge diagonally for every symmetric input.

At every step, however,Thus each iterate is orthogonally similar to its predecessor. Similar matrices have the same characteristic polynomial and eigenvalues, including multiplicities, so induction proves that every iterate has exactly the original eigenvalues. Orthogonal similarity also preserves symmetry.