Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A x \le c^+ x^\top D x $$
for any $x \in \mathbb{R}^n$, where $c^+, c^- > 0.$ The matrices $A$ and $D$ can be diagonalized, that is $$ A = V\Lambda\_A V^\top, \quad D = W\Lambda\_D W^\top $$
where $V$ and $W$ contain the eigenvectors of $A$ and $D$, and $\Lambda\_A$ and $\Lambda\_D$ are diagonal matrices containing the respective eigenvalues.
Based on the Reighleigh quotient, it should follow that
$$ cond(D^{-1}A) \le \frac{c^+}{c^-},$$
thus $c^+$ and $c^-$ upper and lower bounds for the range of the eigenvalues of $D^{-1}A.$


**Now my question is**: For $0 < \alpha \le 1,$ does
$$ (c^-)^\alpha x^\top D^\alpha x \le x^\top A^\alpha x \le (c^+)^\alpha x^\top D^\alpha x $$
hold ? Here, $A^\alpha := V\Lambda\_A^\alpha V^\top,$ and $D^\alpha := W\Lambda\_D^\alpha W^\top,$ where $\Lambda\_A^\alpha, \Lambda\_D^\alpha$ can be computed by taking the power $\alpha$ of each diagonal entry.



prove spectral equivalence bounds for fractional power of matrices
$\newcommand\C{\mathbb C}\newcommand\R{\mathbb R}\newcommand\al{\alpha}$Yes, this follows by Loewner's theorem on monotone matrix functions (see e.g. [Theorem 1.6](https://link.springer.com/book/10.1007/978-3-030-22422-6)), which in particular implies the following:


Let $M\_n$ denote the set of all analytic functions $f\colon\C\setminus(-\infty,0]\to\C$ such that $f((0,\infty))\subseteq\R$ and
$$A\le B\implies f(A)\le f(B)$$
for all $n\times n$ positive-definite matrices $A$ and $B$, where $A\le B$ means that $B-A$ is positive semidefinite.
Then $f\in M\_n$ for all natural $n$ if
$$\Im z>0\implies \Im f(z)>0.$$


The above conditions on $f$ hold if $f(z)=z^\alpha$ for $\alpha\in(0,1]$ and all $z\in\C\setminus(-\infty,0]$.


So, your desired result immediately follows.


Even more immediately, your desired result follows from [Theorem 4.1](https://link.springer.com/book/10.1007/978-3-030-22422-6), which in turn follows from the identity
$$x^\al=\frac{\sin\pi\al}\pi\int\_0^\infty w^{\al-1}x(x+w)^{-1}\,dw$$
for real $x>0$, since in this identity $x$ can be replaced by any positive-definite matrix $A$ (see [formula (4.5)](https://link.springer.com/book/10.1007/978-3-030-22422-6)), and the monotonicity of $A(A+w)^{-1}=I-w(A+w)^{-1}$ in $A$ for $w>0$ is easy to check (say, by differentiation).


