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Example
Created on 2024-06-20Asked by Owen Campbell (Solvelet student)
Find the eigenvectors corresponding to the eigenvalue λ=3 \lambda = 3 for the matrix A=[2−1−12] A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix} .

Solution

To find the eigenvectors corresponding to the eigenvalue λ=3 \lambda = 3 for the matrix A=[2−1 −12] A = \begin{bmatrix} 2 & -1 \ -1 & 2 \end{bmatrix} :Step 1: Subtract λ \lambda times the identity matrix from A A : A−λI=[2−3−1 −12−3]=[−1−1 −1−1]. A - \lambda I = \begin{bmatrix} 2-3 & -1 \ -1 & 2-3 \end{bmatrix} = \begin{bmatrix} -1 & -1 \ -1 & -1 \end{bmatrix}. Step 2: Solve the equation (A−λI)v=0 (A - \lambda I) \mathbf{v} = \mathbf{0} to find the eigenvector v \mathbf{v} : [−1−1 −1−1]v=0. \begin{bmatrix} -1 & -1 \ -1 & -1 \end{bmatrix} \mathbf{v} = \mathbf{0}. Step 3: Find the solution space for the system of equations: {−v1−v2=0 −v1−v2=0. \begin{cases} -v_1 - v_2 = 0 \ -v_1 - v_2 = 0 \end{cases}. The system has infinitely many solutions, with v2 v_2 being a free variable. Step 4: Choose a value for v2 v_2 (e.g., v2=1 v_2 = 1 ) and solve for v1 v_1 : −v1−1=0  ⟹  v1=−1. -v_1 - 1 = 0 \implies v_1 = -1. Step 5: Conclusion. The eigenvector corresponding to the eigenvalue λ=3 \lambda = 3 for the matrix A A is v=[−1 1] \mathbf{v} = \begin{bmatrix} -1 \ 1 \end{bmatrix} . Solved on Solvelet with Basic AI Model
Some of the related questions asked by Daniel Taylor on Solvelet
1. Find the eigenvectors corresponding to the eigenvalue λ=2 \lambda = 2 for the matrix A=[3−142] A = \begin{bmatrix} 3 & -1 \\ 4 & 2 \end{bmatrix} .2. Determine whether the vector v=⟨1,2⟩ \mathbf{v} = \langle 1, 2 \rangle is an eigenvector of the matrix B=[2132] B = \begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix} corresponding to an eigenvalue of 4 4 .
DefinitionNon-zero vectors that point to lines along which matrices when multiplied, pull, squash, or spin, are known as Eigenvectors. An eigenvector v with a corresponding eigenvalue λ for a certain matrix A will solve the equation Av=λv, and eigenvectors serve as tools in various fields of mathematics (as they provide an algebraic way to analyze stability and dynamic or bifurcation behavior of a system with an equlibrium point) in the solution of numerous issues(code) in linear algebra, differential equations and physics. They are useful for diagonalizing matrices and in solving systems of linear equations. For instance, the eigenvectors of a 2x2 matrix [ac​bd​] correspond to the solutions of the equation [a−λc​bd−λ​]v=0, where λ is the eigenvalue.
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