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Jacobi algorithm

Symmetric eigenvalue computation by repeated plane rotations

Under construction. This chapter is still being developed and may change as additional Jacobi strategies and examples are added.

How can repeated rotations diagonalize a symmetric matrix?

For a real symmetric matrix, the Jacobi algorithm repeatedly applies orthogonal similarity transformations that diagonalize selected 2×2 pivot blocks. The chapter derives the rotation algebraically, shows how it acts inside a larger matrix, explains why previously created zeros can reappear and connects convergence to the decrease of off-diagonal energy.

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Key ideas

The Jacobi method attacks the symmetric eigenproblem by repeatedly rotating selected coordinate pairs while preserving similarity and symmetry.

  • A Jacobi step has the form A′ = G A Gᵀ, so A′ is similar to A and has the same eigenvalues
  • For a symmetric 2×2 pivot block, the rotation angle is chosen so the selected off-diagonal pair becomes zero
  • The same 2×2 rotation can be embedded into any pair of coordinates of an n×n matrix
  • A later similarity transformation can fill a previously created zero back in, so Jacobi does not proceed by permanent triangular elimination
  • Each nonzero Jacobi rotation decreases the off-diagonal energy by twice the square of the selected pivot
  • Accumulating the orthogonal transformations recovers the eigenvectors as the matrix approaches diagonal form

The current chapter builds the algebraic and geometric foundation for the larger preprocessing and parallelization questions studied in the companion tridiagonalization chapter.

Why can a zero created by Jacobi reappear later?

A Jacobi rotation changes two rows and the corresponding two columns. A later rotation involving one of those indices mixes entries again, so an earlier zero can be filled back in. Convergence therefore comes from the overall decrease of off-diagonal energy, not from preserving each zero once it is created.

Related chapters

  • Tridiagonalization & Jacobi algorithm studies whether Householder tridiagonalization can improve the subsequent Jacobi computation and tests the effect experimentally
  • Eigenvectors and eigenvalues develops the underlying eigenvalue, eigenvector and diagonalization ideas used by the Jacobi method
  • Givens rotations uses coordinate-plane rotations for QR elimination and provides a useful comparison with Jacobi plane rotations
  • Householder reflection develops the reflection used for the symmetric tridiagonalization preprocessing studied in the companion chapter

Chapter contents

The chapter is available as a PDF. Page links below are best-effort: most browsers support them but some viewers may ignore the page hint.

Topic Pages
Similarity transformation and eigenvalue computation 1
Orthogonal matrix G that satisfies (A′)₂,₁ = 0 2–5
Two ways to solve the symmetric 2×2 eigenproblem: homogeneous systems and Jacobi rotation 5–7
How one 2×2 Jacobi rotation acts inside an n×n matrix 8–12
Why not systematically zero all subdiagonal entries? 12–13
Why does the Jacobi algorithm converge? 14–16
Single Jacobi rotation in 2D 17–19
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