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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.

An orthogonal similarity transformation is $H=Q^{T}AQ$ with $Q^{T}Q=I$. An upper Hessenberg matrix has $H_{ij}=0$ whenever $i>j+1$; its first subdiagonal may be nonzero.

At step $k=1,\dots ,m-2$, choose a Householder reflector $Q_k=\mathrm{diag}(I_k,{\hat{Q}}_k)$ whose trailing block sends the current column segment in rows $k+1,\dots ,m$ to a multiple of its first coordinate. The left multiplication zeros positions $(k+2,k),\dots ,(m,k)$. Apply the matching right multiplication to retain similarity: $A\leftarrow Q_k^{T}A{Q}_k$. Since $Q_k$ acts only on rows and columns $k+1,\dots ,m$, earlier Hessenberg zeros are preserved. Finally $Q=Q_1\cdots Q_{m-2}$ gives $H=Q^{T}AQ$.

Each reflector is applied through a rank-one update, costing $O(m^2)$ rather than a full matrix multiplication. Across $m-2$ steps, the Hessenberg reduction costs $O(m^3)$.

Solving the $m$ unrelated shifted systems by separate Gaussian elimination costs $\boxed{\frac{2}{3}{m}^4+O(m^3)}$ floating-point operations.

Instead compute $H=Q^{T}AQ$ once in $O(m^3)$ work. For each shift, solveThe shift changes only the diagonal, so $H-{\lambda }_{j}I$ remains an upper Hessenberg matrix. In each elimination column only the entry immediately below the pivot needs removal. With partial pivoting the only possible nonzero candidate rows are the current row and the next, and each update touches $O(m)$ entries. Factorization therefore costs $O(m^2)$ per shift; the triangular solve and both orthogonal-vector transformations also cost $O(m^2)$ each.

Thus $m$ shifted solves plus the initial Hessenberg reduction cost $\boxed{O(m^3)}$ in total. Pivoting is still needed in general: nonsingularity excludes a singular system but does not by itself guarantee that every unpivoted elimination pivot is nonzero.