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The factorization produced by the Arnoldi iteration, with orthonormal $Q_{k+1}$ and an upper Hessenberg matrix ${\tilde{H}}_k$. Multiplying by $Q_k^T$ identifies its leading square block as $H_k=Q_k^{T}A{Q}_k$.

Multiplying the Arnoldi decomposition by $Q_k^T$ shows that the leading square block isIt is an upper Hessenberg matrix by the Arnoldi iteration. Because $A=A^T$, it is also a symmetric matrix. Entries below the first subdiagonal vanish by the Hessenberg structure, and their reflected entries above the first superdiagonal vanish by symmetry. ConsequentlyThe extra last row has only its final entry possibly nonzero. With the usual choice of positive normalization factors, the individual column relations are the three-term Lanczos algorithm recurrencewhere the first term is omitted for $j=1$. This reduction is an exact-arithmetic statement; round-off error can affect the orthogonality in an implementation.