Introduction to Singular Value Decomposition Calculator:
Singular Value Decomposition Calculator is an online tool that helps you to get the solution of singular value decomposition problems in a few seconds. The svd calculator is a very helpful tool for all because it is a very lengthy or difficult process when you do manual calculations.
There is a high chance of making mistakes in manual calculation solutions because so many steps are involved or you may be stuck in evaluation. That is why we make this calculator that gives you instant results in steps for easy understanding.
What is Singular Value Decomposition?
Singular Value Decomposition (SVD) is a complex method in linear algebra where the matrix factorization method decomposes any given matrix A into three constituent matrices U, VT, and Σ matrix. Singular value decomposition formula is:
$$ A \;=\; UΣV^T $$
Where:
- U is an m×m orthogonal matrix.
- Σ is the singular value in an m×n diagonal matrix with real numbers(non-negative) on the diagonal.
- V is the transpose of the orthogonal matrix in n×n.
Working Method Behind Singular Value Calculator:
SVD decomposition calculator use the easiest method to calculate the singular value decomposition value problem that will be understandable to all.
It has an advanced algorithm that makes it a faster tool that gives solutions even for a complex singular value problem. Here are the given calculation steps of singular value decomposition with the help of an example.
Example: Find SVD - Singular Value Decomposition,
$$ \left[ \begin{matrix} 4 & 1 \\ 3 & -5 \\ \end{matrix} \right] $$
Step-by-Step Calculation:
Given matrix A is,
$$ A \;=\; \left[ \begin{matrix} 4 & 1 \\ 3 & -5 \\ \end{matrix} \right] $$
Step 1: Take matrix A and find the transpose of matrix A. Then take a dot product of A.AT such as:
$$ A^T \;=\; \left[ \begin{matrix} 4 & 1 \\ 3 & -5 \\ \end{matrix} \right]^T \;=\; \left[ \begin{matrix} 4 & 3 \\ 1 & -5 \\ \end{matrix} \right] $$
$$ A \times (A^T) \;=\; \left[ \begin{matrix} 4 & 1 \\ 3 & -5 \\ \end{matrix} \right] \times \left[ \begin{matrix} 4 & 3 \\ 1 & -5 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 4 \times 4 + 1 \times 1 & 4 \times 3 + 1 \times -5 \\ 3 \times 4 - 5 \times 1 & 3 \times 3 - 5 \times -5 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 16 + 1 & 12 - 5 \\ 12 - 5 & 9 + 25 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 17 & 7 \\ 7 & 34 \\ \end{matrix} \right] $$
Step 2: Find the eigenvalues using A.AT solution such as:
$$ |A.A^T - λI | \;=\; 0 $$
$$ \left| \begin{matrix} (17 - \lambda) & 7 \\ 7 & (34 - \lambda) \\ \end{matrix} \right| \;=\; 0 $$
$$ (17 - \lambda) \times (34 - \lambda) - 7 \times 7 \;=\; 0 $$
$$ (578 - 51\lambda + \lambda^2) - 49 \;=\; 0 $$
$$ (\lambda^2 - 51\lambda + 529) \;=\; 0 $$
$$ (\lambda - 14.4886422272)(\lambda - 36.5113577728) \;=\; 0 $$
$$ (\lambda - 14.4886422272) \;=\; 0 \;or \; (\lambda - 36.5113577728) \;=\; 0 $$
The eigenvalues of the matrix A . A’ are given by,
λ = 14.4886422272, 36.5113577728
Step 3: Find the eigenvector with the help of λ values one by one.
Example: Eigenvectors for λ = 36.5113577728,
$$ A . A’ - λI \;=\; \left[ \begin{matrix} 17 & 7 \\ 7 & 34 \\ \end{matrix} \right] - 36.511357773 \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 17 & 7 \\ 7 & 34 \\ \end{matrix} \right] - \left[ \begin{matrix} 36.511357773 & 0 \\ 0 & 36.511357773 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} -19.511357773 & 7 \\ 7 & -2.511357773 \\ \end{matrix} \right] $$
Now reduce this matrix,
$$ R_1 \leftarrow R_1 \div -19.511357773 $$
$$ \left[ \begin{matrix} 1 & -0.3587653961 \\ 7 & -2.511357773 \\ \end{matrix} \right] $$
$$ R_2 \leftarrow R_2 - 7 \times R_1 $$
$$ =\; \left[ \begin{matrix} 1 & -0.3587653961 \\ 0 & 0 \\ \end{matrix} \right] $$
The system associated with the eigenvalue λ = 36.5113577728.
$$ (A . A’ -36.5113577731) \left[ \begin{matrix} x_1 \\ x_2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 1 & -0.3587653961 \\ 0 & 0 \\ \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 \\ 0 \\ \end{matrix} \right] $$
$$ \rightarrow x_1 - 0.3587653961 x_2 \;=\; 0 $$
$$ \rightarrow x_1 \;=\; 0.3587653961 x_2 $$
$$ v \left[ \begin{matrix} 0.3587653961 x_2 \\ x_2 \\ \end{matrix} \right] $$
Let x2 = 1,
$$ v_1 \;=\; \left[ \begin{matrix} 0.3587653961 \\ 1 \\ \end{matrix} \right] $$
Eigenvectors for λ = 14.4886422272,
$$ A . A’ - λI \;=\; \left[ \begin{matrix} 17 & 7 \\ 7 & 34 \\ \end{matrix} \right] - 14.488642227 \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 17 & 7 \\ 7 & 34 \\ \end{matrix} \right] - \left[ \begin{matrix} 14.488642227 & 0 \\ 0 & 14.488642227 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 2.511357773 & 7 \\ 7 & 19.511357773 \\ \end{matrix} \right] $$
Now reduce the matrix, interchanging rows R1 ↔ R2
$$ \left[ \begin{matrix} 7 & 19.511357773 \\ 2.511357773 & 7 \\ \end{matrix} \right] $$
$$ R_1 \leftarrow R_1 \div 7 $$
$$ =\; \left[ \begin{matrix} 1 & 2.7873368147 \\ 2.511357773 & 7 \\ \end{matrix} \right] $$
$$ R_2 \rightarrow R_2 - 2.511357773 \times R_1 $$
$$ =\; \left[ \begin{matrix} 1 & 2.7873368247 \\ 0 & 0 \\ \end{matrix} \right] $$
The system is associated with the eigenvalue λ = 14.4886422272.
$$ (A . A’ - 14.4886422271) \left[ \begin{matrix} x_1 \\ x_2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 1 & 2.7873368247 \\ 0 & 0 \\ \end{matrix} \right]\; \left[ \begin{matrix} x_1 \\ x_2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 0 & 0 \\ \end{matrix} \right] $$
$$ \rightarrow x_1 + 2.7873368247x_2 \;=\; 0 $$
$$ \rightarrow x_1 \;=\; -2.7873368247x_2 $$
Eigenvectors corresponding to the eigenvalue λ = 14.4886422272 is,
$$ v \;=\; \left[ \begin{matrix} -2.7873368247x_2 \\ x_2 \\ \end{matrix} \right] $$
Let x2 = 1,
$$ v_2 \;=\; \left[ \begin{matrix} -2.7873368247 \\ 1 \\ \end{matrix} \right] $$
These are the eigen vector v1,v2 get from the given matrix,
Step 4: Now find the length of the given eigenvector v1 and v2.
For Eigenvalue - 1 (0.3587653961, 1), Length L = √0.35876539612 + 12 = 1.0624088711.
For Eigenvector - 2 (-2.7873368247, 1), Length L = √(-2.7873368247)2 + 12 = 2.9612913694.
Step 5: Compute v1/‖v1‖ and v2/‖v2‖ to normalize the given eigenvector as
$$ u_1 \;=\; \frac{v_1}{‖v1‖}\; \; \; u_2 \;=\; \frac{v2}{‖v2‖} $$
$$ u_1 \;=\; \left( \frac{0.3587653961}{1.0624088711} , \frac{1}{1.0624088711} \right) \;=\; (0.3376905124, 0.9412572007) $$
$$ u_2 \;=\; \left( \frac{-2.7873368247}{2.9612913694} , \frac{1}{2.9612913694} \right) \;=\; (-0.9412572007, 0.3376905124) $$
Step 6: Construct a matrix U with the help of u1 and u2 values,
$$ U \;=\; [u1, u2] \;=\; \left[ \begin{matrix} 0.3376905124 & -0.9412572007 \\ 0.9412572007 & 0.3376905124 \\ \end{matrix} \right] $$
Step 7: Construct a matrix Σ after taking the square root of eigenvalues λ1 and λ2.
$$ \sum \;=\; \left[ \begin{matrix} \sqrt{36.5113577728} & 0 \\ 0 & \sqrt{14.4886422272} \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} 6.0424628896 & 0 \\ 0 & 3.8063949121 \\ \end{matrix} \right] $$
Step 8: Lastly, to make a matrix V, you need to put a value in this formula, where you find all the values except the value of σ which is equal to σ1=√λ1, σ2=√λ2 and so on.
V is found using formula vi = 1/σi AT . ui
$$ V \;=\; \left[ \begin{matrix} 0.6908662458 & -0.7229825935 \\ -0.7229825935 & -0.6908662457 \\ \end{matrix} \right] $$
Step 9: Now you have matrix U,V and Σ, put all the values in the formula to get the solution of singular value decomposition.
$$ A \;=\; U \sum V^T $$
$$ U \times \sum \;=\; \left[ \begin{matrix} 0.33769051244 & -0.94125720067 \\ 0.94125720067 & 0.33769051244 \\ \end{matrix} \right] \times \left[ \begin{matrix} 6.04246288965 & 0 \\ 0 & 3.80639491215 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 0.33769061244 \times 6.04246288965 - 0.94125720067 \times 0 & 0.33769051244 \times 0 - 0.94125720067 \times 3.80639491215 \\ 0.94125720067 \times 6.04246288965 + 0.33769051244 \times 0 & 0.94125720067 \times 0 + 0.33769051244 \times 3.80639491215 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 2.04048238961 + 0 & 0 - 3.58279661965 \\ 5.68751170466 + 0 & 0 + 1.28538344843 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 2.04048238961 & -3.58279661965 \\ 5.68751170466 & 1.28538344843 \\ \end{matrix} \right] $$
$$ (U \times \sum) \times (V^T) \;=\; \left[ \begin{matrix} 2.04048238961 & -3.58279661965 \\ 5.68751170466 & 1.28638344843 \\ \end{matrix} \right] \times \left[ \begin{matrix} 0.69086624576 & -0.7229825935 \\ -0.72298259353 & -0.69086624572 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 2.04048238961 \times 0.69086624576 - 3.58279661965 \times -0.72298259353 \\ 2.04048238961 \times -0.7229825935 - 3.58279661965 \times -0.69086624572 \\ 5.68751170466 \times 0.69086624576 + 1.28538344843 \times -0.72298259353 & 5.68751170466 \times -0.7229825935 + 1.28538334483 \times -0.69086624572 \\ \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 1.40970040805 + 2.59029959271 & -1.47523325003 + 2.4752332498 \\ 3.92930985912 - 0.92930985923 & -4.1119719628 - 0.88802803733 \end{matrix} \right] $$
$$ =\; \left[ \begin{matrix} 4.00000000021 & 0.99999999977 \\ 2.99999999989 & -5.00000000013 \\ \end{matrix} \right] $$
How to Use Singular Values of a Matrix Calculator?
Singular Value Decomposition Calculator has a simple design that helps you to solve the given matrix singular decomposition value question instantly. You just need to put your problem in this calculator and follow some important steps. These steps are:
- Choose the size of the matrix from the given list.
- Enter the elements of your matrix to get the solution of SVD in the input fields.
- Review your given input value to get the correct result of the singular value decomposition question.
- Click on the Calculate button to get the result of the given singular value decomposition problems.
- You can use the load example for a solution to check the workings behind our svd calculator.
- Click the “Recalculate” button for the calculation of more examples of singular value decomposition in solution.
Output Get from Singular Value Calculator:
SVD decomposition calculator provides you with a solution as per your input problem when you click on the calculate button. It may include as:
In the Result Box:
Click on the result button so you get the solution to your singular value decomposition question.
Steps Box:
When you click on the steps option, you get the result of singular value decomposition questions in a step-by-step process.
Useful Features of Singular Values of a Matrix Calculator:
(SVD) Singular Value Decomposition Calculator has many useful features that you get when you use it to solve singular value decompostion problems to get its solution. Our tool only gets the input value and it provides a solution without taking external help. These features are:
- It is a reliable tool as it always provides you with accurate solutions to given singular value decomposition problems.
- SVD calculator is an efficient tool that provides solutions to the given singular value decomposition problems in a few seconds.
- It is a learning tool that provides you with complete knowledge about the concept of singular value decomposition very easily on online platforms.
- A singular value calculator is a handy tool that solves various types of singular value decomposition problems quickly without manual calculation.
- It is a free tool that allows you to use it for calculation without charging anything.
- SVD decomposition calculator is an easy-to-use tool, anyone or even a beginner can easily use it for the solution of singular value decomposition problems.
