Introduction to Singular Value Calculator:
Singular value calculator is an online source that helps you to find the singular values of a square matrix. Our tool helps you to determine the singular value of various kinds of square matrices even it is in n order in a few seconds.
It is a very useful tool for students, and professionals who want to solve singular value problems without doing manual calculation because it is used in matrix decompositions, solving linear equations, and data analysis in mathematics.
What is the Singular Value of a Matrix?
Singular values are a set of real or complex values that can be found from the eigenvalues of a square matrix. It is an important method in linear algebra. If you have an m×n of matrix A, the singular values are denoted by σi as i=1,2,…,n. Singular values are used to provide in-depth knowledge insights about the structure and properties of matrices.
Calculation Process of Singular Values Calculator:
The singular values of a matrix calculator are used to calculate the singular values solution from matrices (real or complex) because it has an up-to-date algorithm in its software. To calculate singular values of matrix, various steps are used, which are:
Step 1: Determine the given matrix A and its order. This matrix can be real or complex.
Step 2: Take the transpose of the matrix A*= AT where AT=A is the conjugate transpose A in an n×n matrix.
Step 3: Compute the matrix A* .A
Step 4: Find the eigenvalues λi from the A* .A matrix.
Step 5: For the singular values σi of matrix A, take the square roots of the eigenvalues of A^*. Thus, σi=√λi for i=1,2,…, min(m,n).
Step 6: Arrange the singular values σi in non-increasing order σ1 ≥ σ2 ≥ … ≥σ min(m,n) ≥ 0.
Solved Example of a Singular Value:
Let's take an example of a singular value problem to understand how the singular values of a matrix calculator solve such problems.
Example: Consider the following matrix
$$ A \;=\; \left( \begin{matrix} 3 & 1 \\ 1 & 2 \\ \end{matrix} \right) $$
Solution:
Identify the given matrix A and take the transpose of A such as AT=A*
$$ A* \;=\; A^T \;=\; \left( \begin{matrix} 3 & 1 \\ 1 & 2 \\ \end{matrix} \right) $$
Multiply A* .A,
$$ A* A \;=\; \left(\begin{matrix} 3 & 1 \\ 1 & 2 \\ \end{matrix} \right) \left( \begin{matrix} 3 & 1 \\ 1 & 2 \\ \end{matrix} \right) \;=\; \left( \begin{matrix} 10 & 5 \\ 5 & 5 \\ \end{matrix} \right) $$
Find the eigenvalue from the equation det(λ - AI),
$$ det(A* A - λI) \;=\; 0 $$
$$ det \left( \left( \begin{matrix} 10 - λ & 5 \\ 5 & 5 - λ \\ \end{matrix} \right) \right) \;=\; 0 $$
Solve the determination to find the value of λ.
$$ (10 - λ)(5 - λ) - 5 \times 5 \;=\; 0 $$
$$ 50 - 10λ - 5λ + λ^2 - 25 \;=\; 0 $$
$$ λ^2 - 15λ + 25 \;=\; 0 $$
After solving this equation with a quadratic formula we get
$$ λ \;=\; 13,\; λ \;=\; 2 $$
To get the singular value, take the square root of λ=13, λ=2 such as:
$$ σ_1 \;=\; \sqrt{13} \approx 3.61 $$
$$ σ_2 \;=\; \sqrt{2} \approx 1.41 $$
Therefore, the singular values of the matrix A are approximately 3.61 and 1.41.
How to Use the Singular Value Matrix Calculator?
Singular value of a matrix calculator has a simple design that makes it easy for you how to use it for the evaluation of square matrix for singular value, follow our instructions that are given as:
- Choose the size of the matrix from the given field to find the singular value.
- Enter the element of a square matrix to find the singular value in the input field.
- Review the given matrix before hitting the calculate button to start the evaluation process in the singular values of matrix calculator.
- Click the “Calculate” button to get the result of your given singular value problem.
- If you are trying our singular value calculator for the first time then you can use the load example to learn more about this method.
- Click on the “Recalculate” button to get a new page for finding more example solutions of singular value problms.
Outcome From Singular Values Calculator:
Singular values of a matrix calculator give you the solution from a given matrix for a singular value problem when you add the input into it. It included as:
- Result Option:
When you click on the result option the singular value matrix calculator gives you a solution to the square matrix problem to find a singular value.
- Possible Steps:
When you click on it, this option will provide you with a solution where all the calculations of singular value process in steps.
Advantages of Singular Value of a Matrix Calculator:
Singular values of matrix calculator provide you with many advantages that help you to calculate square matrix singular value problems and give you solutions without any trouble. These advantages are:
- Singular value calculator is a free tool so you can use it for free to find the singular value problem solutions without paying.
- It is an adaptable tool that can manage various types of matrices to calculate singular values of matrix.
- Our singular values calculator helps you to get conceptual clarity for the singular value process when you use it for practice by solving more examples.
- It saves the time that you consume on the calculation of the singular value problems.
- The singular values of a matrix calculator is a reliable tool that provides you with accurate solutions whenever you use it to calculate the singular value without any man-made error after evaluation.
- Singular value matrix calculator provides the solution without imposing any condition of signup after two to three uses which means you can use it multiple times.
