Consider space $\mathcal{S}=\mathbb{R}^{n\times n}$. Let $\mathbf{X},\mathbf{Y}\in\mathcal{S}$. The inner product between $\mathbf{X},\mathbf{Y}$ is defined as (https://en.wikipedia.org/wiki/Frobenius_inner_product)
\begin{equation} <\mathbf{X},\mathbf{Y}> = trace(\mathbf{Y}^{T}\mathbf{X}^{})=\sum_{i=1}^{n}\sum_{j=1}^{n} \mathbf{X}_{i,j}\mathbf{Y}_{i,j} \end{equation}
In $\mathbb{R}^2,\mathbb{R}^3$, "Inner Products" give us a notion of "angle" between the two vectors i-e
\begin{equation} <\mathbf{x},\mathbf{y}> = \|\mathbf{x}\|\|\mathbf{y}\| cos(\theta) \end{equation}
I am wondering how to extend this geometric interpretation to matrices? What does it mean for two matrices to be "aligned". Does it mean their singular vectors are aligned?