Diagonalization Calculator

Using the Diagonalization Calculator

This guide will walk you through the steps necessary to use the Diagonalization Calculator, a tool designed to compute the diagonal, eigenvector, and inverse matrices based on provided eigenvalues and eigenvectors for a 2×2 matrix.

Step 1: Input Matrix Size

Begin by entering the matrix size you wish to work with. In this calculator, the only valid input is 2 since it is designed for 2×2 matrices. Ensure that your input is an integer and follows the guidelines given. The field will require a number between 2 and 5, but the current setup only practically applies to a 2×2 matrix scenario.

Step 2: Enter Eigenvalues

• First Eigenvalue (λ₁): Input the first eigenvalue in the designated field. This should be a numeric value, and this field is required for the calculations to proceed.
• Second Eigenvalue (λ₂): Similar to the first, enter the second eigenvalue. Ensure this field is filled out as it is integral to forming the diagonal matrix.

Step 3: Enter Eigenvectors

• First Eigenvector: Enter the components of the first eigenvector. Start with the x-component labeled as First Eigenvector x-component (v₁ₓ), followed by the y-component labeled as First Eigenvector y-component (v₁ᵧ). These components must both be provided.
• Second Eigenvector: Similarly, fill in the components for the second eigenvector. Enter the x-component in the field labeled as Second Eigenvector x-component (v₂ₓ) and the y-component in Second Eigenvector y-component (v₂ᵧ).

Step 4: Review the Results

Once all inputs are provided, the calculator will automatically compute several matrices and values:

• Diagonal Matrix (D): This matrix will display with the eigenvalues on its diagonal. The result reflects the entered eigenvalues in positions D₁₁ and D₂₂, formatted to two decimal places.
• Eigenvector Matrix (P): Displays how the eigenvectors are structured in a 2×2 matrix form. Review the layout and ensure it matches the input components, formatted to two decimal places.
• Inverse of Eigenvector Matrix (P⁻¹): This matrix highlights the inverse of the eigenvector matrix and utilizes the determinant in its calculation. Check the values for any errors if the determinant input doesn’t allow the computation. Values will appear with four decimal precision.
• Determinant of Eigenvector Matrix: Displays the determinant of the eigenvector matrix. This value is crucial for ensuring the invertibility of the matrix and is detailed to four decimal places.
• Original Matrix (A = PDP⁻¹): Reflects the reconstructed original matrix wherein the product of the diagonal matrix and inverse matrix yields the input matrix. The calculations are executed to four decimal points for accuracy check.

Use this guide effectively to navigate the input fields and understand the results displayed by the Diagonalization Calculator. Ensure precision in entering the eigenvalues and eigenvectors to obtain correct and meaningful outputs.