Problem Set 7: Eigenvalues and Eigenvectors of 2\times2 Matrices

Problem Set 7: Eigenvalues and Eigenvectors of \(2\times2\) Matrices#

Instructions#

Find the eigenvalues and eigenvectors for each matrix, then normalize them to produce unit eigenvectors. Verify your work using the eig function from SciPy’s linalg module.

Part A: Real Eigenvalues#

Problem 1. Find the eigenvalues and eigenvectors of

\[\begin{split}A = \begin{bmatrix} 5 & 2 \\ 2 & 2 \end{bmatrix}\end{split}\]

Problem 2. Find the eigenvalues and eigenvectors of

\[\begin{split}B = \begin{bmatrix} 3 & 1 \\ 0 & 3 \end{bmatrix}\end{split}\]

Part B: Irrational Eigenvalues#

Problem 3. Find the eigenvalues and eigenvectors of

\[\begin{split}C = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}\end{split}\]

Problem 4. Find the eigenvalues and eigenvectors of

\[\begin{split}D = \begin{bmatrix} 1 & 2 \\ 1 & 1 \end{bmatrix}\end{split}\]

Part C: Complex Conjugate Eigenvalues#

Problem 5. Find the eigenvalues and eigenvectors of

\[\begin{split}E = \begin{bmatrix} 1 & -2 \\ 2 & 1 \end{bmatrix}\end{split}\]

Problem 6. Find the eigenvalues and eigenvectors of

\[\begin{split}F = \begin{bmatrix} 0 & 1 \\ -4 & 0 \end{bmatrix}\end{split}\]

Problem 7. Find the eigenvalues and eigenvectors of

\[\begin{split}G = \begin{bmatrix} 2 & -5 \\ 1 & -2 \end{bmatrix}\end{split}\]

Part D: Linear Combinations of Eigenvectors#

Problem 8. Consider the matrix

\[\begin{split}H = \begin{bmatrix} 4 & 1 \\ 2 & 3 \end{bmatrix}\end{split}\]

(a) Find the eigenvalues and eigenvectors of \(H\).

(b) Write the vector \(\mathbf{v} = \begin{bmatrix} 7 \\ 8 \end{bmatrix}\) as a linear combination of the eigenvectors of \(H\).

(c) Verify your answer by computing the linear combination explicitly.

Problem 9. Consider the matrix

\[\begin{split}K = \begin{bmatrix} 1 & 4 \\ 1 & 1 \end{bmatrix}\end{split}\]

(a) Find the eigenvalues and eigenvectors of \(K\).

(b) Write the vector \(\mathbf{w} = \begin{bmatrix} 6 \\ 3 \end{bmatrix}\) as a linear combination of the eigenvectors of \(K\).

Bonus Challenge#

Problem 10. A matrix \(M\) has eigenvalues \(\lambda_1 = 3\) and \(\lambda_2 = -1\) with corresponding eigenvectors \(\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \end{bmatrix}\) and \(\mathbf{v}_2 = \begin{bmatrix} 1 \\ -1 \end{bmatrix}\).

(a) Reconstruct the matrix \(M\).

(b) Verify that your matrix has the given eigenvalues and eigenvectors.

Tips#

  • Remember: \(\det(A - \lambda I) = 0\) gives the characteristic equation

  • For 2×2 matrices: if \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), then \(\det(A - \lambda I) = \lambda^2 - \text{tr}(A)\lambda + \det(A)\)

  • To find eigenvectors, solve \((A - \lambda I)\mathbf{v} = \mathbf{0}\)

  • Complex eigenvalues come in conjugate pairs for real matrices

  • Eigenvectors can be scaled by any nonzero constant

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