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Eigenvalues and Eigenvectors Calculator

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Example
Created on 2024-06-20Asked by Benjamin Rivera (Solvelet student)
Find the eigenvalues and eigenvectors of the matrix A=[3−1−13] A = \begin{bmatrix} 3 & -1 \\ -1 & 3 \end{bmatrix} .

Solution

To find the eigenvalues and eigenvectors of the matrix A=[3−1 −13] A = \begin{bmatrix} 3 & -1 \ -1 & 3 \end{bmatrix} :Step 1: Write down the characteristic equation: det(A−λI)=0, \text{det}(A - \lambda I) = 0, where I I is the identity matrix. Step 2: Substitute the values of A A into the characteristic equation: det([3−λ−1 −13−λ])=(3−λ)2−(−1)(−1)=λ2−6λ+8=0. \text{det}\left(\begin{bmatrix} 3-\lambda & -1 \ -1 & 3-\lambda \end{bmatrix}\right) = (3-\lambda)^2 - (-1)(-1) = \lambda^2 - 6\lambda + 8 = 0. Step 3: Solve the characteristic equation to find the eigenvalues: (λ−4)(λ−2)=0  ⟹  λ1=4,λ2=2. (\lambda - 4)(\lambda - 2) = 0 \implies \lambda_1 = 4, \quad \lambda_2 = 2. Step 4: For each eigenvalue, find the corresponding eigenvector v \mathbf{v} by solving (A−λI)v=0 (A - \lambda I) \mathbf{v} = \mathbf{0} : For λ1=4 \lambda_1 = 4 : [−1−1 −1−1]v1=0  ⟹  v1=[1 −1]. \begin{bmatrix} -1 & -1 \ -1 & -1 \end{bmatrix} \mathbf{v}_1 = \mathbf{0} \implies \mathbf{v}_1 = \begin{bmatrix} 1 \ -1 \end{bmatrix}. For λ2=2 \lambda_2 = 2 : [1−1 −11]v2=0  ⟹  v2=[1 1]. \begin{bmatrix} 1 & -1 \ -1 & 1 \end{bmatrix} \mathbf{v}_2 = \mathbf{0} \implies \mathbf{v}_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix}. Step 5: Conclusion. The eigenvalues of A A are λ1=4 \lambda_1 = 4 and λ2=2 \lambda_2 = 2 , with corresponding eigenvectors v1=[1 −1] \mathbf{v}_1 = \begin{bmatrix} 1 \ -1 \end{bmatrix} and v2=[1 1] \mathbf{v}_2 = \begin{bmatrix} 1 \ 1 \end{bmatrix} , respectively. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Aria Adams on Solvelet
1. Find the eigenvectors and eigenvalues of the matrix A=[2−143] A = \begin{bmatrix} 2 & -1 \\ 4 & 3 \end{bmatrix} .2. Determine whether the vector v=⟨1,2⟩ \mathbf{v} = \langle 1, 2 \rangle is an eigenvector of the matrix B=[3124] B = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} corresponding to an eigenvalue of 5 5 .
DefinitionThe eigenvectors and their associated eigenvalues are their definition : * Eigenvectors & EigenValues are one of the most important concepts in Linear Algebra. Scalar quantities that only mutually multiply the original eigenvectors when the matrix multiplier is a matrix are called Eigenvalues, while nonzero vectors which are new direction in the eigenvector of the square matrix when multiplied by the matrices is called eigenvectors. They are used for system behavior analysis, vibration mode calculation, and solving linear equations. The tools furnish information about the algebraic and geometric properties of linear transformations. Example: The eigenvalues and eigenvectors of a matrix A are in λ det(A−λ1I) = 0, where I is the identity matrix.
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