ConvertOnlineMatrix 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.
- Enter the matrix size for matrix A.
- Based on the given matrix size, a matrix of input fields appears. Enter the matrix elements.
- 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 \( \frac{5 - \sqrt{33}}{2} \).