Householder bidiagonalization ($B=PAQ,P,Q\text{ orthogonal}$)

Alternating right and left Householder transformations reduce a real square matrix to a bidiagonal matrix. For a lower-bidiagonal result, first clear row $j$ beyond column $j$ using a right transformation on trailing columns. Then clear column $j$ below row $j+1$ using a left transformation on trailing rows. Previously completed rows and columns are preserved. Since $B^{T}B=Q^T{A}^{T}AQ$ is a symmetric tridiagonal matrix, its eigenvalues are the squares of the singular values of $A$. This explains the reduction structurally; explicitly forming $A^{T}A$ is unnecessary and can worsen numerical conditioning.