svd

Singular value decomposition

Introduction

Let $A$ be a real $m\times n$ matrix. Then, there exist orthogonal matrices $U\in\mathbb{R}^{m\times m}$ and $V\in\mathbb{R}^{n\times n}$ such that

where $V'$ is the transpose of $V$. The matrix $\Sigma=\text{diag}(\sigma_1,\sigma_2,\ldots,\sigma_p)$, where $p=\min(m,n)$. The real numbers $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_p\geq 0$ are called the singular values of $A$.

If $A\in\mathbb{C}^{m\times n}$ is complex, there exist unitary matrices $U\in\mathbb{C}^{m\times m}$ and $V\in\mathbb{C}^{n\times n}$ such that

where $V'$ is the conjugate transpose of $V$. The real numbers $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_p\geq 0$ are called the singular values of $A$.

Thin SVD

The following results hold for real $A\in\mathbb{R}^{m\times n}$. Similar results hold for complex $A\in\mathbb{C}^{m\times n}$ as well and hence we omit the discussion here.

If $A\in\mathbb{R}^{m\times n}$ with $m> n$, then

where

• $U_1\in\mathbb{R}^{m\times n}$ contains the first $n$ columns of $U$.
• $\Sigma_1\in\mathbb{R}^{n\times n}$ contains the first $n$ columns and first $n$ rows of $\Sigma$.

If $A\in\mathbb{R}^{m\times n}$ with $m\leq n$, then

where

• $V_1\in\mathbb{R}^{n\times m}$ contains the first $m$ columns of $V$.
• $\Sigma_1\in\mathbb{R}^{m\times m}$ contains the first $m$ columns and first $m$ rows of $\Sigma$.

Usages

s = svd(A)

s = svd(A,0)

s = svd(A,'econ')

• A is a real or complex $m\times n$ matrix. It should not contain Inf nor NaN.
• It returns the column vector of singular values, i.e., $[\sigma_1,\sigma_2,\ldots,\sigma_p]'$, where $p=\min(m,n)$ and $\sigma_1\geq\sigma_2\geq\cdots\geq\sigma_p\geq 0$.

[U,S,V] = svd(A)

• A is a real or complex $m\times n$ matrix. It should not contain Inf nor NaN.
• It returns the matrices $U$, $\Sigma$ and $V$ in Equation (1) or (2).

[U,S,V] = svd(A,0)

• A is a real or complex $m\times n$ matrix. It should not contain Inf nor NaN.
• When $m>n$, it returns $U_1$, $\Sigma_1$ and $V$ of Equation (3)
• When $m\leq n$, it returns $U$, $\Sigma$ and $V$ of Equation (1) or (2).

[U,S,V] = svd(A, 'econ')

• A is a real or complex $m\times n$ matrix. It should not contain Inf nor NaN.
• When $m>n$, it returns $U_1$, $\Sigma_1$ and $V$ of Equation (3).
• When $m=n$, it returns $U$, $\Sigma$ and $V$ of Equation (1) or (2).
• When $m< n$, it returns $U$, $\Sigma_1$ and $V_1$ of Equation (4).