# Singular Value Decomposition Calculator

## Find the SVD of a matrix step by step

The calculator will find the singular value decomposition (SVD) of the given matrix, with steps shown.

Related calculator: Pseudoinverse Calculator

The SVD calculator is a reliable companion for all your singular value decomposition tasks. This tool allows you to determine a matrix's singular values swiftly and efficiently, making your numerical linear algebra operations smoother. Thanks to its intuitive interface, you're only a click away from performing sophisticated, singular value decomposition calculations.

## How to Use the Singular Value Decomposition Calculator?

### Input

Begin by entering the matrix elements into the provided input fields. Our calculator supports matrices of different sizes.

### Calculation

Once inputting your matrix, click the "Calculate" button. The calculator will immediately start working on the singular value decomposition.

### Result

After a short moment, the calculator will display the $$$U$$$, $$$\Sigma$$$, and $$$V$$$ matrices, which are the primary components of the SVD. The singular values of the matrix, i.e., the diagonal elements of the $$$\Sigma$$$ matrix, will also be provided.

## What Is Singular Value Decomposition?

Singular Value Decomposition (SVD) is a fundamental procedure in linear algebra. It breaks down a matrix into several constituent parts, allowing for the detailed examination of its structure and properties. This powerful method offers valuable insights that find use in multiple fields, such as data science, machine learning, and image processing.

At its core, SVD is a method of factorizing a matrix. Given a matrix $$$A$$$ of dimension $$$m\times n$$$, we can decompose it into three separate matrices: $$$U$$$, $$$\Sigma$$$, and $$$V$$$. This is represented mathematically as:

$$A=U\Sigma V^*$$Here:

- $$$U$$$ is an $$$m\times m$$$ real or complex unitary matrix.
- $$$\Sigma$$$ is an $$$m\times n$$$ rectangular diagonal matrix.
- $$$V^*$$$ (the conjugate transpose of $$$V$$$) is an $$$n\times n$$$ real or complex unitary matrix

The diagonal entries of $$$\Sigma$$$ (denoted as $$$\sigma_i$$$) are known as the singular values of the matrix $$$A$$$. They are always non-negative and are usually listed in descending order. The columns of $$$U$$$ and $$$V^*$$$ are called the left-singular and right-singular vectors of $$$A$$$, respectively.

For instance, consider a 2x2 matrix $$$A$$$:

$$A=\left[\begin{array}{cc}3&1\\0&4\end{array}\right]$$Using the SVD calculator, we can decompose $$$A$$$ into $$$U$$$, $$$\Sigma$$$, and $$$V^*$$$:

$$U=\left[\begin{array}{cc}\frac{\sqrt{5}}{5}&-\frac{2\sqrt{5}}{5}\\\frac{2\sqrt{5}}{5}&\frac{\sqrt{5}}{5}\end{array}\right]$$$$\Sigma=\left[\begin{array}{cc}3\sqrt{2}&0\\0&2\sqrt{2}\end{array}\right]$$$$V^*=\left[\begin{array}{cc}\frac{\sqrt{10}}{10}&\frac{3\sqrt{10}}{10}\\-\frac{3\sqrt{10}}{10}&\frac{\sqrt{10}}{10}\end{array}\right]$$The singular values of $$$A$$$ are thus the diagonal elements of $$$\Sigma$$$, i.e., $$$3\sqrt{2}$$$ and $$$2\sqrt{2}$$$.

The power of SVD lies in its ability to reveal the inherent structure of the matrix. By examining the singular values and vectors, we can understand the matrix's behavior, rank, null space, and other properties. In practical applications, SVD is used in solving systems of linear equations, computing matrix inverses, data compression, principal component analysis (PCA), and more.

## Why Choose Our Singular Value Decomposition Calculator?

### Accuracy

Our SVD calculator is designed to provide precise results. It carefully performs the decomposition process to ensure you receive accurate $$$U$$$, $$$\Sigma$$$, and $$$V$$$ matrices every time.

### User-Friendly Interface

Our calculator is easy to use with an intuitive design. You have to input your matrix, and with a single click, you'll get the singular value decomposition.

### Speed

Time is valuable, and we respect that. Our calculator swiftly performs the computations, providing you with results in just a few moments.

### Flexibility

The SVD Calculator can handle matrices of different sizes. Whether it's a simple 2x2 matrix or a larger, more complex one, the calculator can manage it.

### Educational Value

Besides providing quick results, our calculator can help you better understand the singular value decomposition process. By seeing how a matrix is decomposed into $$$U$$$, $$$\Sigma$$$, and $$$V$$$, you can better understand this fundamental concept in linear algebra.

### FAQ

#### What is the use of Singular Value Decomposition (SVD)?

SVD is a powerful tool that finds applications in many fields. It's extensively used in data science for principal component analysis (PCA), in machine learning for pattern recognition, and in image processing for image compression and enhancement, among others.

#### What is Singular Value Decomposition?

Singular Value Decomposition, or SVD, is a method of factorizing a matrix into three separate matrices: $$$U$$$, $$$\Sigma$$$, and $$$V$$$. The diagonal entries of $$$\Sigma$$$ are known as the singular values of the matrix. SVD is widely used in various fields, such as image processing, machine learning, and data science.

#### Why does the order of singular values matter?

The order of singular values, typically arranged in descending order, is significant as it reflects the importance of the corresponding singular vectors in representing the original matrix's structure. The larger the singular value, the more significant its corresponding singular vector is.

#### What's the difference between eigenvalues and singular values?

While both eigenvalues and singular values provide valuable insights into a matrix's properties, they are different concepts. Eigenvalues come from the eigendecomposition of a square matrix, whereas singular values come from the singular value decomposition of any (not necessarily square) matrix. Also, eigenvalues can be negative or complex, whereas singular values are always non-negative.