Similarity transformation — where it appears
Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.
Two Laplacians of one graph
The combinatorial Laplacian D − A and the normalised one, which conjugates it by the inverse square roots of the degrees, are built from the same object, are not similar to each other, and answer different questions. On a graph whose degrees are equal they coincide. On one whose degrees span an order of magnitude their second eigenvalues are sixteen times apart.
The algorithm the libraries actually run
Factorise, multiply the factors back in the other order, repeat. That description is complete and correct and produces something nobody would use — on a matrix with eigenvalues +1 and −1 it does not converge at all, and the subdiagonal entry does not move by so much as a rounding error.
The form that makes it affordable
One Householder reduction, done once, turns every subsequent iteration of the eigenvalue algorithm from cubic to quadratic cost. It changes no answer at all, which is why it is easy to describe as an optimisation and wrong to.
The form a real matrix can reach
A real matrix with complex eigenvalues has no real triangular form, and the reason is one line — a real triangular matrix has a real diagonal, and a similarity does not move the spectrum. What it has instead is triangular except for one two-by-two block per conjugate pair, and the count is decided by the matrix rather than by where the iteration stopped.
A certificate written in coordinates
Every one-sided reduced model of the convection–diffusion system was stable, because its poles cannot leave the numerical range and the range sat in the left half plane. Rescale the state by a diagonal matrix and nothing about the system changes — not its eigenvalues, not its transfer function, not one pole of any two-sided model — but the numerical range moves. At Péclet 40 a rescaling with condition number 260 pushes its edge to +6.6 and ten placements of nineteen return unstable one-sided models. Projecting in the inner product of a Lyapunov solution puts the certificate back in any coordinates.
Balanced is not symmetric
The Sylvester–Kac matrix is made of small integers, so a double holds it exactly, and its eigenvalues are the integers from −(n − 1) to n − 1 in steps of two. Every digit an eigensolver loses on it is therefore the solver's own, and it loses them at exactly the rate first-order perturbation theory predicts: the median error is half the prediction across 1,568 eigenvalues. Balancing, the preprocessing libraries apply for this kind of matrix, divides every condition number by about sixty and leaves their growth untouched, and at order 112 the unbalanced solver returns eighteen complex eigenvalues for a spectrum of integers.
The second factor was the last row
Wilkinson's polynomial in Chebyshev coefficients loses its roots exponentially from the colleague matrix, and two factors accounted for the loss: the roots' own conditioning, and the size of the matrix's last row against the rest of it. A diagonal similarity by powers of two — balancing, which changes no eigenvalue and rounds nothing — removes the second factor entirely: the balanced error sits within a factor of two and a half of the conditioning at every degree from fourteen to forty, three and a half thousand times better than unbalanced at forty. What is left is the conditioning, still exponential. Balancing the monomial companion matrix takes its norm down by forty-five orders and gains nothing, because there the error was never the matrix's.
Named alongside it
The objects these essays reach for when they reach for this one.
DeflationJacobi's eigenvalue methodNon-normalityOrthogonalityThe QR algorithmShiftsTridiagonalAlgebraic connectivityBackward errorChebyshev basisCompanion formComplex conjugate pair