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The matrix 2-norm is the operator norm induced by the Euclidean norm:It equals the largest singular value of $A$ and is invariant under multiplication by orthogonal or unitary matrices.

For a full-column-rank $n\times k$ matrix $V$, extend the thin QR decomposition $V=Q_1{R}_1$ to an orthogonal matrix $Q=(Q_1{Q}_2)$. Thenso $Q^T$ maps the column space of $V$ onto the first $k$ coordinate directions. Successive Householder transformations construct the same reduction without first forming a full orthogonal basis.

For a proposal $Y=AX+Z$, with isotropic $Z\sim N(0,I)$ and an independent random orthogonal matrix $A$, invariance of the distribution of $A$ under transpose makes the proposal distribution symmetric. Indeed $|y-Ax|=|x-A^{T}y|$, and averaging the isotropic Gaussian density over an inversion-invariant law yields $q(x,y)=q(y,x)$. The Metropolis–Hastings algorithm therefore accepts with the target density ratio alone.