About Singular Value Decomposition(SVD) calculator
This is a free online Singular Value Decomposition(SVD) calculator with complete, detailed, step-by-step description of solutions, that performs operations with matrices up to 99x99 in size with matrix elements of this type: decimal numbers, fractions, complex numbers, variables.
To start the calculation, you need to first enter the size of the matrix in the input field that you can find from the very top of the screen, also there you can choose the desired method of calculation.
A little below you will find a matrix window in which you need to enter matrix elements using the keyboard. The matrix control panel is also located here, which simplifies work with matrices and contains the following control elements:
- The first element allows you to expand the matrix window. This can be especially useful in cases where you need to perform calculations with very large matrices that do not fit completely. If the matrix is still not visible after expanding the window, you can change the scale of the matrix using the + / - buttons;
- The second element performs the function of copying the matrix input to the memory buffer. This can be useful in cases where you often use the same matrix for calculations, or if you need to move matrices between operations;
- And the last element inserts the previously copied matrix, which allows you to speed up the process of entering the matrix to just a few clicks, instead of doing it manually;
And further down you will find a toolbar that allows you to customize the calculator and make it easier to work with it. It is visually divided into three parts, each of which is responsible for the following functionality:
- The first allows you to select the number format when the solution result is displayed. Also, here you can turn off comments to the solution of the problem if you have already understood how to solve this problem, and you use the calculator to speed up or check your own calculations. Or you can turn off the step-by-step solution entirely if you only need the result of the solution;
- The second contains buttons that allow you to change the type of the matrix input field, erase its elements or the entire matrix, and the largest button with an equal sign, which will take you to the screen with the solution of the problem. All these buttons are duplicated by keys on the keyboard. To find out which key on the keyboard to press, simply hover over one of the buttons and a tooltip will appear with the name of the key. You can also use the arrow keys on your keyboard to move the cursor between matrix input fields;
- And the last one allows you to choose the number of digits after the decimal point for rounding non-integer numbers. Also, here you can immediately see an example of how rounded fractions will look;
What is the Singular Value Decomposition(SVD) of a matrix?
Singular Value Decomposition(SVD) is the factorization of a given real or complex matrix into three matrices, one of which is an n x n complex unitary matrix, the second matrix is an n x m rectangular diagonal matrix with singular values(non-negative real numbers) on the diagonal, and the third matrix is an m x m conjugate transpose complex unitary matrix. The product of an n x n unitary matrix by an n x m rectangular diagonal matrix and an m x m conjugate transpose complex unitary matrix should give the original matrix.
How to perform the Singular Value Decomposition(SVD) of a matrix?
We need to find the first Hermitian matrix of the original matrix by multiplying the original matrix by its transposed matrix. Then we need to find the second Hermitian matrix of the original matrix by multiplying the transposed original matrix by the original matrix. After that, we need to calculate the eigenvalues and eigenvectors of the first Hermitian matrix. Now we need to calculate the singular values by taking the square root of each positive eigenvalue of the first Hermitian matrix. This will allow us to compose a rectangular diagonal matrix by placing the singular values on the main diagonal and filling all other elements of the matrix with zeros. Also at this step we can find the n x n complex unitary matrix by normalizing the eigenvectors of the first Hermitian matrix and placing them as the columns of the n x n complex unitary matrix. After that, we need to find the eigenvectors of the second Hermitian matrix, normalize them and place them as columns of the m x m complex unitary matrix. And now it remains only to find conjugate transpose matrix of the m x m complex unitary matrix.
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