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Section 2: Problems

University-level exam questions for Spectral Theory & Matrix Decompositions.

Eigenvalues and Eigenvectors​

Problem 1.1​

Find the eigenvalues and eigenvectors of: A = [[3, 1], [1, 3]]

Difficulty: Easy

Problem 1.2​

Prove that similar matrices have the same eigenvalues.

Difficulty: Medium

Problem 1.3​

If A is invertible, show that the eigenvalues of A⁻¹ are 1/λᵢ where λᵢ are eigenvalues of A.

Difficulty: Medium

Diagonalization​

Problem 2.1​

Diagonalize the matrix A = [[2, 1], [1, 2]] and use it to compute A¹⁰.

Difficulty: Medium

Problem 2.2​

Solve the difference equation xₖ₊₁ = Axₖ with x₀ = [1, 0] where A = [[0.8, 0.3], [0.2, 0.7]].

Difficulty: Hard

Matrix Exponentials​

Problem 3.1​

Compute eᴬᵗ for A = [[0, 1], [-1, 0]].

Difficulty: Hard

Problem 3.2​

Solve the differential equation dx/dt = Ax with x(0) = [1, 0] where A = [[2, 1], [1, 2]].

Difficulty: Hard

SVD and Positive Definiteness​

Problem 4.1​

Compute the SVD of A = [[1, 1], [0, 1], [1, 0]].

Difficulty: Medium

Problem 4.2​

Determine if the following matrix is positive definite: A = [[2, -1, 0], [-1, 2, -1], [0, -1, 2]]

Difficulty: Medium

Problem 4.3​

Find the best rank-1 approximation to A = [[4, 0], [3, -5]] using SVD.

Difficulty: Hard

Challenge Problems​

Problem 5.1​

Prove the spectral theorem for real symmetric matrices.

Difficulty: Very Hard

Problem 5.2​

Show that every positive definite matrix can be written as A = LLᵀ for some lower triangular L (Cholesky decomposition).

Difficulty: Very Hard

Solutions​

Solutions are available in the implementation file with verification code.