Orthogonal similarity ($B=Q^{T}AQ,Q^{T}Q=I$)

An orthogonal matrix similarity changes coordinates by an orthogonal matrix. It preserves eigenvalues, singular values and Euclidean geometry; it also preserves symmetry when applied to a symmetric matrix. A sequence of Householder transformations can use this freedom to produce a Hessenberg reduction, while a symmetric matrix can be reduced further to tridiagonal form.