For the purpose of this article, the abstract tensor <math>\mathcal{A}</math> is assumed to be given in coordinates with respect to some basis as a [[Tensor#As multidimensional arrays|M-way array]], also denoted by <math>\mathcal{A}\in\mathbb{C}^{I_1 \times I_2 \cdots \times \cdots I_m \cdots\times I_M}</math>, where ''M'' is the number of modes and the order of the tensor. <math>\mathbb{C}</math> is the complex numbers and it includes both the real numbers <math>\mathbb{R}</math> and the pure imaginary numbers.
Let <math>\mathcal{A}_{[m]}\in\mathbb{C}^{I_m \times (I_1 I_2 \cdots I_{m-1} I_{m+1} \cdots I_M)}</math> denote the [[Tensor reshaping#Mode-m Flattening / Mode-m Matrixization|standard mode-''m'' flattening]] of <math>\mathcal{A}</math>, so that the left index of <math>\mathcal{A}_{[m]}</math> corresponds to the <math>m</math>'th index <math>\mathcal{A}</math> and the right index of <math>\mathcal{A}_{[m]}</math> corresponds to all other indices of <math>\mathcal{A}</math> combined. Let <math>{\bf U}_m \in \mathbb{C}^{I_m \times I_m}</math>be a [[unitary matrix]] containing a basis of the left singular vectors of the <math>\mathcal{A}_{[m]}</math> such that the ''j''th column <math>\mathbf{u}_j</math> of <math>{\bf U}_m</math> corresponds to the ''j''th largest singular value of <math>\mathcal{A}_{[m]}</math>. Observe that the '''mode/factor matrix''' <math>{\bf U}_m</math> does not depend on the particular on the specific definition of the mode ''m'' flattening. By the properties of the [[multilinear multiplication]], we have<math display="block">\begin{array}{rcl}
