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The singular value decomposition of $B$ is $B=U\Sigma V^{†}$, where $U,V$ are unitary and $\Sigma =\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_n)$, ${\sigma }_1\ge \cdots \ge {\sigma }_n>0$ since $B$ is invertible. Its right singular vector $v_j$ and left singular vector $u_j$ are the corresponding unit columns of $V$ and $U$, withEquivalently $B^{†}B{v}_j={\sigma }_j^2{v}_j$ and $B{B}^{†}{u}_j={\sigma }_j^2{u}_j$. The positive numbers ${\sigma }_j$ are the singular values. Degenerate singular subspaces allow any orthonormal choice of paired vectors.

For $q(t)=B{q}_0$, the energy ratio is the Rayleigh quotient of the Hermitian positive-definite matrix $B^{†}B$:Expanding $q_0$ in the orthonormal right singular vectors makes this a weighted average of ${\sigma }_j^2$. HenceAny nonzero vector in the largest right-singular subspace is optimal if ${\sigma }_1$ is repeated. Its amplified state is proportional to the paired left singular vector. For a non-normal matrix $L$, this optimal initial direction need not be an eigenvector of $L$, explaining transient growth from non-normal modes.

On each left singular vector $u_j$, use $A^{\ast }{u}_j={\sigma }_j{v}_j$ and $(I-A^{\ast }A)v_j=(1-{\sigma }_j^2)v_j$. The finite geometric series givesThus with $\alpha =1/n$, the dimensionless Landweber spectral filter isThe full inverse-coefficient convention instead uses $[1-(1-{\sigma }^2{)}^{1/\alpha }]/\sigma$. With general step $\tau$, replace ${\sigma }^2$ inside the power by $\tau {\sigma }^2$.

In the normalized problem, $0\le f_n(\sigma )\le \min (n{\sigma }^2,1)$, so the inverse coefficient is at most $\min (n\sigma ,1/\sigma )\le \sqrt{n}$. This verifies directly why finite iteration is stable and why admitting progressively smaller singular values eventually amplifies noise.

A unit right singular vector of a matrix or compact operator $A$ is an eigenvector of $A^{\ast }A$ with eigenvalue ${\sigma }_j^2$. For ${\sigma }_j>0$, $u_j=A{v}_j/{\sigma }_j$ is the corresponding left singular vector. The largest singular value is the operator norm of $A$, attained on its corresponding right singular vectors.