# Derivative of eigenvalues of a symmetric tridiagonal matrix built via the Lanczos-Arnoldi scheme

Suppose $$\mathbf{A}(\mu)$$ being a symmetric positive definite matrix of dimension $$n$$ where its elements depend parametrically on the real parameter $$\mu$$.

Suppose now to build the orthonormal basis of the Krylov subspace from an initial normalized guess $$\mathbf{x}_0$$ by performing $$m$$ iterations of the Lanczos-Arnoldi algorithm. This orthonormal basis is gathered in the unitary matrix $$\mathbf{V}$$, dimension ($$n,m$$), and transforms $$\mathbf{A}$$ into a tridiagonal symmetric matrix $$\mathbf{T}$$ of dimension $$m$$ which can be finally diagonalized by a unitary transformation $$\mathbf{U}$$ as shown below where $$\boldsymbol{\Lambda}$$ denotes the final diagonal matrix. $$$$\begin{split} \mathbf{T}&=\mathbf{V}^T\mathbf{AV}\\ \boldsymbol{\Lambda}&=\mathbf{U}^T\mathbf{TU}\end{split}$$$$ We are now interested in the derivative of $$\boldsymbol{\Lambda}$$ with respect to the parameter $$\mu$$: this very problem was already discussed in a previous question (Derivative of eigenvectors of an Hermitian matrix) and the result is reported below. $$$$\frac{d\boldsymbol{\Lambda}}{d\mu}=\mathbf{U}^T\frac{d\mathbf{T}}{d\mu}\mathbf{U}$$$$ In our case, however, $$\mathbf{T}$$ is obtained via the trasfomation $$\mathbf{V}$$ which is the orthonormal basis of the Krylov subspace mentioned.

The question is therefore, can I compute the derivative of $$\mathbf{T}$$ by simply differentiating $$\mathbf{A}(\mu)$$ (equation below), i.e. similarly to the case for $$\boldsymbol{\Lambda}$$ where we ignored the derivatives of the transformation matrix, or should I consider differentiating $$\mathbf{V}$$ as well? $$$$\frac{d\mathbf{T}}{d\mu}=\mathbf{V}^T\frac{d\mathbf{A}}{d\mu}\mathbf{V}$$$$

• no, in the equation for $d\Lambda/d\mu$ it is used that the matrix $\Lambda$ is diagonal, so there is no analogous equation for $dT/d\mu$. Apr 21, 2021 at 17:44
• If we write explicitly $\boldsymbol{\Lambda}=\mathbf{U}^T\mathbf{V}^T\mathbf{A}\mathbf{VU}$, we can gather the two unitary transformations into $\mathbf{W}=\mathbf{VU}$, in this case could we then differentiate $\mathbf{\Lambda}$ as follows? $$\frac{d\boldsymbol{\Lambda}}{d\mu}=\mathbf{W}^T\frac{d\mathbf{A}}{d\mu}\mathbf{W}$$ Is this valid even in the case of $\mathbf{W}$ having dimension $(m,n)$ and thus not being the full transformation containing all the $n$ eivengectors or $\mathbf{A}$? Apr 22, 2021 at 5:58

You decompose the real symmetric matrix $$A$$ as $$A=W\Lambda W^T$$, with $$W$$ the orthogonal matrix of eigenvectors and $$\Lambda={\rm diag}\,(\lambda_1,\lambda_2,\ldots)$$ the diagonal matrix of eigenvalues. Consider one eigenvalue $$\lambda_k$$ and the associated eigenvector $$\psi$$ with elements $$\psi_i=W_{ik}$$.
By construction, the eigenvalue equals the inner product $$\lambda_k=(\psi , A\psi)=(A\psi , \psi),$$ because $$(\psi,\psi)=1$$. Now take the derivative with respect to $$\mu$$, denoted by a prime: $$\lambda'_k=(\psi',A\psi)+(A\psi,\psi')+(\psi,A'\psi)=$$ $$\qquad=\lambda_k(\psi',\psi)+\lambda_k(\psi,\psi')+(\psi,A'\psi)$$ $$\qquad=\lambda_k\frac{d}{d\mu}(\psi,\psi)+(\psi,A'\psi)$$ $$\qquad=0+(\psi,A'\psi).$$ So you see, the derivative of the wave functions drops out because of the normalization.
• Dear Carlo, than you very much you your kind explanation. I think I get you point. So, as long as $\mathbf{W}$ contains the eigenvectors of $\mathbf{A}$, I can differentiate $\boldsymbol{\Lambda}$ without caring of the "eigenvectors' response", i.e. $\mathbf{W}$. What made me dubious was the fact that, by working in a Krylov subspace ($m<n$), the final m eigenvalues of $\boldsymbol{\Lambda}$ are only approximations of the actual eigenv. of $\mathbf{A}$, but as long as the eigenvalue problem $\mathbf{AW}=\mathbf{W}\boldsymbol{\Lambda}$ is solved, the Hellmann Feynman Th. can still be applied. Apr 22, 2021 at 7:15
• In fact, as far as I remember from the old Quantum Chemistry courses, the Hellmann-Feynman theorem is valid both for exact and approximate wavefunctions, as long as the latter are solved variationally, and this could be transferred to the case I reported where we are approximating the exact eigenvalues of $\mathbf{A}$ in a "variational fashion" since we are solving an eigenvalue problem. Apr 22, 2021 at 7:23