# Higher order singular value decomposition projection matrices

Given an $N$-th order tensor $\mathcal{A} \in \mathbb{R}^{I_1\times I_2\times\ldots\times I_N}$ we can find an approximation of $\mathcal{A}$, $\hat{\mathcal{A}}$ by means of Higher Order Singular value Decomposition (HOSVD). That is,

$$\hat{\mathcal{A}}=\mathcal{Y}\times_1\mathbf{U}_1\times_2 \mathbf{U}_2 \ldots \times_N \mathbf{U}_N$$

where $\mathcal{Y}$ is a core tensor, $\times_i$ stands for mode-$i$ product (Tensor matrix multiplication) and matrices $\mathbf{U}_d|_{d=1}^N$ are mentioned as orthogonal projection matrices.

Q1: Matrices $\mathbf{U}_d|_{d=1}^N$ are generally rectangular, not square. However, an orthogonal matrix is generally square. How can this contradiction be explained?

Q2: Similarly a projection matrix is by definition square. How $\mathbf{U}_d|_{d=1}^N$ are projection matrices even though they are rectangular?

Q3: What matrices $\mathbf{U}_d|_{d=1}^N$ project and where? What is the physical interpretation of the term projection in HOSVD?