Equations or areas where $(AA^T)^x$ or $(A^TA)^x$ are used as applications

Let $$A$$ be square or rectangular and $$x\in \mathbb{R}$$. Can you point me to equations/areas out there where $$(AA^T)^x$$ or $$(A^TA)^x$$ or their eigenvalues are used as applications? e.g. we find them in the polar decomposition when $$x=\frac{1}{2}$$.

I'm particular about cases when $$x\ne 0, \frac{1}{2}, 1$$. Thanks in advance.

$$A$$ may also be taken to be complex i.e. $$(AA^*)^x$$ or $$(A^*A)^x$$.

• what does $A^x$ mean for real $x$? – Pink Panther Mar 17 at 14:14
• Would you be willing to use a formula in terms of the eigenvalues of $A^TA$? There's also a formula using a power series. – Omnomnomnom Mar 17 at 14:36
• @Omnomnomnom. Yes, I will consider their eigenvalues too. I have edited the post to include that $A$ can also be rectangular. – Kay Mar 17 at 15:47
• @Omnomnomnom, please expantiate on the formula using a power series, and that which considers the eigenvalues. Thanks in advance. – Kay Mar 17 at 18:01

$$A^TA$$ is a symmetric and positive semidefinite matrix. By the spectral theorem, there exists an orthogonal $$U$$ such that $$U^T(A^TA)U$$ is diagonal, that is $$U^T(A^TA)U = D = \pmatrix{\lambda_1 \\ & \ddots \\ && \lambda_n}$$ where the $$\lambda_k$$ are the eigenvalues of $$A^TA$$ (i.e. the singular values of $$A$$). Because $$A^TA$$ is positive semidefinite, the eigenvalues $$\lambda_k$$ are non-negative, that is $$\lambda_k \geq 0$$ for all $$k$$.
We may compute $$(A^TA)^x = [U D U^T]^x = UD^x U^T = U \pmatrix{\lambda_1^x \\ & \ddots \\ && \lambda_n^x}U^T$$ Because $$\lambda_k\geq 0$$ for all $$k$$, $$\lambda_k^x$$ makes sense for all $$x > 0$$. If we also know that $$A^TA$$ is invertible (i.e. that $$A$$ has full column rank), then we have $$\lambda_k > 0$$ which means that $$\lambda_k^x$$ makes sense for all $$x \in \Bbb R$$.