This tool helps you find the diagonal form of a given square matrix for easier interpretation and analysis.

### How to Use the Diagonalization Calculator

To use this calculator, enter the matrix you want to diagonalize in the input field in the format as described below:

- Comma separated rows, semi-colon separated elements.
- For example, for a 2×2 matrix, you would enter: “1,2; 3,4”.

After entering the matrix, click the ‘Calculate’ button. The result will display two matrices P and T. P is an orthogonal matrix, and T is a diagonal matrix such that P^{T}AP = T.

### Explanation of Calculation

The calculator performs the following steps to diagonalize the input matrix:

- Parsing the input matrix into a 2D array structure.
- Applying matrix decomposition to obtain orthogonal matrix P and diagonal matrix T.

Note: The actual decomposition logic is simplified in this calculator. In a full-fledged application, you would typically employ libraries to handle complex linear algebra operations.

### Limitations

This calculator has the following limitations:

- Handles only numerical matrices.
- Uses simplified decomposition, not suitable for higher-degree polynomials or highly complex diagonalization tasks.
- Assumes the input matrix is a square matrix.

For accurate and complex diagonalization tasks, consider using specialized math libraries.

## Use Cases for This Calculator

### Calculate the diagonal of a rectangle

Find the diagonal of a rectangle by entering the length and width values. The calculator will utilize the Pythagorean theorem to provide you with the accurate diagonal measurement, making it easy to determine the distance from one corner to the opposite corner of the rectangle.

### Determine the diagonal of a square

Easily calculate the diagonal of a square by entering the side length. The calculator will swiftly compute the diagonal length using the Pythagorean theorem, giving you the precise measurement and eliminating the need for manual calculations.

### Find the diagonal of a parallelogram

Quickly find the diagonal of a parallelogram by inputting the base and height values. The calculator will take care of the diagonal calculation for you, saving time and ensuring accuracy in determining the distance between opposite corners of the parallelogram.

### Calculate the hypotenuse of a right triangle

Input the two perpendicular side lengths of a right triangle to compute the hypotenuse. The calculator applies the Pythagorean theorem to determine the hypotenuse, simplifying the process of finding the longest side of the triangle without the need for manual arithmetic.

### Determine the diagonal of a trapezoid

Effortlessly find the diagonal measurement of a trapezoid by entering the bases and height of the shape. Let the calculator do the work for you in calculating the diagonal length accurately, allowing you to focus on other aspects of your project or task.

### Calculate the diagonal of a rhombus

Input the side length of a rhombus to swiftly determine the diagonal length. The calculator will simplify the process of finding the diagonal using the Pythagorean theorem, providing you with the precise measurement for your rhombus shape.

### Determine the diagonal of a kite

Enter the two diagonal lengths of a kite to calculate the remaining diagonal length. Let the calculator handle the diagonal computation for you, ensuring accuracy and efficiency in finding the missing diagonal measurement of the kite shape.

### Find the diagonal of a polygon

Easily calculate the diagonal of a polygon by inputting the number of sides and side length. The calculator will compute the diagonal measurement, simplifying the process of determining the distance between vertices at opposite corners of the polygon.

### Calculate the length of a diagonal brace

Input the horizontal and vertical spans to compute the length of a diagonal brace in a structure. The calculator will swiftly determine the diagonal length, facilitating the accurate installation and support of diagonal braces in construction projects.

### Determine the diagonal measurement in a 3D space

Enter the three-dimensional coordinates to calculate the diagonal length within a 3D space. The calculator will apply the distance formula to find the precise diagonal measurement between two points in space, aiding in spatial calculations and planning.