Let $A(t), A_0$ be a $n\times n$ hermitian complex matrices and consider the following matrix flow

\begin{align} \frac{dA}{dt} &= \left [ C\circ A , A \right ] \\ A(0) &= A_0 \ . \end{align}

Here $\circ$ stands for Hadamard (pointwise) multiplication and the (antisymmetric) matrix $C$ has the form

$$ C=\left(\begin{array}{ccccc} 0 & -1 & -1 & \cdots & -1\\ 1 & 0 & -1 & & \vdots\\ 1 & 1 & 0 & \ddots & -1\\ \vdots & & \ddots & \ddots & -1\\ 1 & 1 & \cdots & 1 & 0 \end{array}\right). $$

I am not 100% sure of the nomenclature Toda vs Morse flow, in any case the flow above or slight variation thereof, has also been considered in a few occasions on MO, for example here. It can be shown that the flow is isospectral and that as $t\to\infty$ , $A(t)$ converges to a diagonal matrix with the eigenvalues of $A_0$ ordered from lowest to largest.

My question is the following:

Is anything known about the convergence rate of $\mathrm{diag}(A(t))$ to the eigenvalues of $A_0$? Can we show that the convergence is bounded by some exponential (and in this case can we estimate the rates)?

If the question above is too complicated I would be equally interested in the convergence rate of the flow with the following modified $C$:

$$ C_1=\left(\begin{array}{ccccc} 0 & -1 & -1 & \cdots & -1\\ 1 & 0 & 0 & 0 & \vdots\\ 1 & 0 & 0 & \ddots & 0\\ \vdots & & \ddots & \ddots & 0\\ 1 & 0 & \cdots & 0 & 0 \end{array}\right). $$

In case of the flow with $C_1$, it can be shown that the matrix element $A_{1,1}(t)$ converges to the lowest eigenvalue of $A_0$, $\lambda_1$.

For the $C_1$-flow, can it be shown that $A_{1,1}(t)$ converges exponentially fast to $\lambda_1$?

It is usually said that the Toda flow is a sort of continuous version of Lanczos diagonalization algorithm. This similarity is even more marked in case of the $C_1$ flow. In fact in the latter case one can show that the "diagonalization" amounts to a (continuous) series of rank-2 unitary rotation (reminiscent of Lanczos algorithm). Since the convergence of the Lanczos algorithm is exponential in the number of iteration, I formulate the conjecture that the convergence of the above flows is also exponential. Is this true? Can it be proven? I couldn't find anything in the literature linked to the MO answers.

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