Firstly, we consider a large data matrix $X\in \mathbb{C}^{n\times m}$:
The column $x_k\in \mathbb{C}^n$ is obtained from simulations or experiments. Here,we always consider the column vectors may also represent the state of a physical system that is evolving in time.The column are often called snapshots and $m$ is the number of snapshots in $X$.
The Singular Value Decomposition allows us to decompose any complex-valued matrix as the product of three other matrices,
where $U\in \mathbb{C}^{n\times n}$ and $V\in\mathbb{C}^{m\times m}$ are unitary matrices. If a square matrix $U$ satisfies $U^{\star}U=UU^{\star}=I$, we call it unitary matrix. Here * denotes the complex conjugate transpose. For real-valued matrices, it is the same as the regular transpose, $X^{\star}=X^T$. So if $U$ and $V$ are real-valued matrices, we call them orthogonal matrices. Since the conclusions of real-valued and complex-valued $U$ and $V$ are parallel, we will only discuss the real-valued case in the following. $\Sigma\in \mathbb{R}^{n\times m}$ is a diagonal matrix, which means real,non-zero entries on thediagonal and zeros off the diagonal.
These column of $U$ have the same shape as a column of $X$. The columns of $U$ describe the eigen of the data in $X$. So in the case of faces, these mean eigenfaces and in the case of flow fields, these mean eigenflow fields. Furthermore, the columns of $U$ are hierarchically arranged in terms of their ability to describe the variance in the column of $X$. In other words, $u_1$ is somehow more important than $u_2$ and so on and so forth.
$U$ gives me a basis, based on which we can represent each column of original data in $X$.Actually, these basis have great properties. Since $U$ is an unitary matrix, it means the column of $U$ are orthonormal. So these column are all orthogonal , have unit length and provide a complete basis for n-dimensional subspace where the column of data matrix lives.
$\Sigma$ is not only a diagnol matrix, but also non-negative and hierarchically ordered matrix. So we have $\sigma_1\geq\sigma_2\geq\sigma_3\geq\cdots\geq\sigma_m\geq0$. They are all non-negative, although some of them could be zero. According to the matrix multiplication, we can find that $\sigma_1$ correspond to the first columns of $U$ and $V$.Since $\sigma_1\geq\sigma_2$, it means that the first columns are somehow important than the second ones when we describe the information of $X$. In other words, the sigular value provides the relative imporatance of these corresponding columns of $U$ and $V$. Finally, we say that the sigular values are ordered by importance.
Here we will start in the case of flow fields. The columns of $U$ will be eigen flows hierarchically organized. We call the first column of $V$ as $v_1$. So $v_1$ would be the time series for how this first eigen mode $u_1$ evolves in this flow.
The matrix $U$ is called left sigular column and the columns are called left singular vectors. $V$ is similar to $U$. The diagonal elements of $\Sigma$ are called sigular values.The rank of X is equal to the number of non-zero sigular value.