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Matrix Eigen ValuesCalculator

Matrix A size: x





Answer




How to use this Matrix Eigen Values Calculator 🤔

Follow these steps to perform Matrix Eigen Values for the given matrices.

  1. Enter the matrix size for matrix A.
  2. Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
  3. As and when you complete entering the matrix elements, Matrix Eigen Values is calculated, and displayed in the answer section.

Matrix Eigenvalues

Eigenvalues are special numbers associated with a matrix that provide insights into its properties and behavior, particularly in transformations. For a given square matrix A, an eigenvalue \( \lambda \) is a scalar such that:

\( A\mathbf{v} = \lambda\mathbf{v} \)

where \( \mathbf{v} \) is the corresponding eigenvector.

To find the eigenvalues of a matrix A, we solve the characteristic equation:

\( \det(A - \lambda I) = 0 \)

where \( I \) is the identity matrix of the same dimension as A.

Example 1

Let's find the eigenvalues of the following 2x2 matrix:

\( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \)

The characteristic equation is:

\( \det\left(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} - \lambda \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\right) = 0 \)

This simplifies to:

\( \det\begin{bmatrix} 1 - \lambda & 2 \\ 3 & 4 - \lambda \end{bmatrix} = 0 \)

Calculating the determinant, we get:

\( (1 - \lambda)(4 - \lambda) - (2 \times 3) = \lambda^2 - 5\lambda - 2 = 0 \)

Solving the quadratic equation \( \lambda^2 - 5\lambda - 2 = 0 \), we find the eigenvalues:

\( \lambda = \frac{5 \pm \sqrt{25 + 8}}{2} = \frac{5 \pm \sqrt{33}}{2} \)

Therefore, the eigenvalues of A are \( \frac{5 + \sqrt{33}}{2} \) and \( \frac{5 - \sqrt{33}}{2} \).