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The spectrum: reading an operator off its eigenvalues

The full set of eigenvalues — the spectrum — is a fingerprint. We'll read trace, determinant, and growth rate straight off it, and confront the question of whether the spectrum even exists over the reals.

What the spectrum is

The spectrum of a finite-dimensional operator is simply the set of its eigenvalues, each a root of the characteristic polynomial. (In infinite dimensions the definition broadens, which you'll meet in the operators track; here, spectrum = eigenvalue set.) Treating the eigenvalues collectively, rather than one at a time, is what lets you summarize an operator at a glance.

Think of the spectrum as the operator's fingerprint: it is invariant under change of basis, since similar matrices A and P^-1 A P share the same characteristic polynomial and hence the same eigenvalues. Two matrices that look completely different on the page can carry the identical spectrum — and the next two sections show how much of A's behavior that single fingerprint already pins down.

Trace and determinant fall out for free

Two invariants of A are encoded in the spectrum. Counting each eigenvalue with its algebraic multiplicity, the trace is the sum of the eigenvalues and the determinant is their product. So the trace and the determinant are nothing but the first and last symmetric functions of the spectrum — Vieta's formulas applied to the characteristic polynomial.

A = [2, -1; 1, 4]

characteristic poly:
  det(A - x I) = (2 - x)(4 - x) - (-1)(1)
               = x^2 - 6x + 9 = (x - 3)^2
  spectrum = {3, 3}   (lambda = 3, algebraic mult 2)

check via Vieta:
  trace(A)       = 2 + 4 = 6  = 3 + 3   = sum of eigenvalues   OK
  det(A)         = 2*4 - (-1)*1 = 9 = 3*3 = product of eigenvalues OK
Trace = sum, determinant = product of eigenvalues (with multiplicity).

Does the spectrum even exist? Over R vs C

A real matrix can have no real eigenvalues at all — a 90-degree rotation [0, -1; 1, 0] has characteristic polynomial x^2 + 1, which has no real roots. The fix is to work over C: by the fundamental theorem of algebra every degree-n polynomial splits, so over C an n x n matrix always has exactly n eigenvalues counted with multiplicity. Real spectra can be empty; complex spectra never are.

One number summarizes the spectrum's size: the spectral radius, the largest |lambda| over the spectrum. It controls long-run behavior — A^k stays bounded exactly when the spectral radius is at most 1 (with care at equality) — and reappears as the convergence rate in Guide 5's power-iteration story.