Jacobi Symmetric Plane Diagonalization

Worked Example

Let us calculate the Jacobi symmetric plane diagonalization for a matrix A = [[2, 1], [1, 4]].

1. Identify the matrix elements: First diagonal a = 2, Second diagonal d = 4, Off diagonal b = 1.
2. Calculate the rotation angle θ:
   θ = 0.5 * atan2(2b, d - a)
   θ = 0.5 * atan2(2(1), 4 - 2) = 0.5 * atan2(2, 2) = 0.5 * (π / 4) ≈ 0.392699
3. Compute the rotation matrix J:
   c = cos(θ) ≈ 0.92388
   s = sin(θ) ≈ 0.38268
   J = [[c, s], [-s, c]]
4. Calculate the rotated matrix J^T A J:
   The new first diagonal a' = c(ac - bs) - s(bc - ds) ≈ 1.585786
   The new second diagonal d' = s(as + bc) + c(bs + dc) ≈ 4.414214
   The new off-diagonal b' = c(as + bc) - s(bs + dc) = 0
5. Verify properties:
   Trace = a' + d' = 1.585786 + 4.414214 = 6 (which equals a + d = 2 + 4)
   Determinant = a' * d' = (1.585786)(4.414214) = 7 (which equals a*d - b*b = 8 - 1)

Experiment with the Controls

You can interact with the diagonalization model at https://riano.app/module/jacobi-symmetric-plane-diagonalization.

1. Set First diagonal to 2, Second diagonal to 2, and Off diagonal to 1.
2. Observe that the rotation radians become exactly π/4 (~0.785398) because the diagonals are equal, meaning the matrix needs a 45-degree rotation to decouple.
3. Notice how the Off-diagonal energy after instantly becomes 0.
4. Set the Off diagonal to 0. The angle becomes 0, and the initial matrix is already diagonal.

Limitations

When implementing the Jacobi rotation using floating-point arithmetic, the annihilated off-diagonal element b' may not be exactly zero but rather a very small number (e.g., 1e-16). Additionally, this module performs only a single local 2x2 rotation; a full Jacobi eigenvalue method for larger matrices requires repeatedly applying such rotations (sweeps) across all off-diagonal pairs until the entire off-diagonal energy falls below a threshold. Finally, the positions of the eigenvalues on the diagonal are not guaranteed to be sorted by magnitude when b ≠ 0.

References

Dongarra, J., et al. "The Jacobi Method". Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, 2000. Provides the foundational formulas for the local 2x2 rotation and demonstrates how it forms the basis of the Jacobi eigenvalue method.

https://www.netlib.org/utk/people/JackDongarra/etemplates/node93.html