Ky Fan Trace Maximum Principle

The Ky Fan Trace Maximum Principle states that for a real symmetric diagonal matrix A with descending eigenvalues, the maximum trace of Q^T*A*Q over orthonormal k-column frames Q is exactly the sum of the k largest eigenvalues.

Worked Example

Consider the "Positive" spectrum with eigenvalues (5, 2, 1) and frame rank k=2.

A = diag(5, 2, 1)
k = 2

Ky Fan maximum = sum(eigenvalues[0..k-1])
               = 5 + 2 = 7

With rotation degrees = 45, the probe Q is:
Q = [
  [cos(45), 0],
  [0, 1],
  [-sin(45), 0]
]

Frame trace = trace(Q^T * A * Q)
            = (5 * cos^2(45) + 1 * sin^2(45)) + (2 * 1)
            = (5 * 0.5 + 1 * 0.5) + 2
            = 3 + 2 = 5.0

Variational gap = 7 - 5.0 = 2.0
Projector trace = 2

Experiment

By adjusting the Spectrum, Frame rank, and Rotation degrees controls in the interactive module, you can observe how the frame trace varies:

Select the Positive option for Spectrum to use the positive spectrum preset.
Set the Frame rank to 2 to choose a 2-column frame.
Change the Rotation degrees to 45 to evaluate the trace at this specific rotation angle.
Observe that the Frame trace shows 5.0, which is less than or equal to the Ky Fan maximum of 7.

Limitations

The interactive visualization uses a finite family of rotations (the first k columns of R_y(theta)) to probe the trace. This serves to illustrate the variational bound and the theorem, but it is not an exhaustive search over every possible orthonormal frame.

References

Primary references for the Ky Fan Trace Maximum Principle:

Ky Fan (1949). "On a Theorem of Weyl Concerning Eigenvalues of Linear Transformations I." Proceedings of the National Academy of Sciences, 35(11), 652–655.PMC1063101

Ky Fan Trace Maximum Principle