Question

Find all eigenvalues and eigenvectors of the given matrix. $$ \left(\begin{array}{cc}{1} & {\sqrt{3}} \\ {\sqrt{3}} & {-1}\end{array}\right) $$

Short answer

A: The eigenvalues can be found by solving the characteristic equation: λ₁ = 0 and λ₂ = 2. The corresponding eigenvectors can be found by solving the system of linear equations: For λ₁ = 0, the eigenvector x₁ = (1/2)(1, -√3)ᵀ For λ₂ = 2, the eigenvector x₂ = (1/2)(1, √3)ᵀ

Step-by-step solution

Write down the given matrix

First, let's write down the given matrix A: $$ A=\left(\begin{array}{cc}{1} & {\sqrt{3}} \\\ {\sqrt{3}} & {-1}\end{array}\right) $$

Find the characteristic equation for the eigenvalues

The characteristic equation can be found by evaluating the determinant of \((A-\lambda I)\), denoted by \(|A-\lambda I|\), where \(\lambda\) is an eigenvalue, and \(I\) is the identity matrix. For given matrix A: $$ |A-\lambda I|=| \begin{pmatrix} 1-\lambda & \sqrt{3} \\ \sqrt{3} & -1 - \lambda \end{pmatrix} | = (1-\lambda)(-1-\lambda)-(\sqrt{3})^2 $$

Solve the characteristic equation

Next, let's solve the characteristic equation: $$ (1-\lambda)(-1-\lambda)-3=-(\lambda^2-2\lambda) $$ Solving the above quadratic equation, we get two eigenvalues: $$ \lambda_1=0, \: \lambda_2=2 $$

Find the eigenvectors for each eigenvalue

For each eigenvalue, let's find the eigenvectors by solving the following system of linear equations \((A-\lambda_i I)x=0\). For \(\lambda_1=0\), \((A-\lambda_1 I)x=0\) becomes: $$ A x = \left(\begin{array}{cc}{1} & {\sqrt{3}} \\\ {\sqrt{3}} & {-1}\end{array}\right) \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} =0 $$ Solving the above system, we get the eigenvector for \(\lambda_1\): $$ x_{1} = \frac{1}{2} \begin{pmatrix} 1 \\ -\sqrt{3} \end{pmatrix} $$ For \(\lambda_2=2\), \((A-\lambda_2 I)x=0\) becomes: $$ (A-\lambda_2 I)x = \left(\begin{array}{cc}{-1} & {\sqrt{3}} \\\ {\sqrt{3}} & {-3}\end{array}\right) \begin{pmatrix} x_3 \\ x_4 \end{pmatrix} =0 $$ Solving this system of equations, we get the eigenvector for \(\lambda_2\): $$ x_{2} = \frac{1}{2} \begin{pmatrix} 1 \\ \sqrt{3} \end{pmatrix} $$

State the final solution

The eigenvalues and corresponding eigenvectors for the given matrix are: $$ \lambda_{1} = 0, \: x_{1} = \frac{1}{2} \begin{pmatrix} 1 \\ -\sqrt{3} \end{pmatrix} $$ and $$ \lambda_{2} = 2, \: x_{2} = \frac{1}{2} \begin{pmatrix} 1 \\ \sqrt{3} \end{pmatrix} $$