Quiz: Eigendecomposition and the Spectral Theorem

1. The Spectral Theorem for real symmetric matrices states that $A \in \mathbb{R}^{n \times n}$ with $A^\top = A$ can be written as $A = Q\Lambda Q^\top$ where:

2. For $A = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$, the eigenvalues are $\lambda_1 = 2$ and $\lambda_2 = 4$. The corresponding normalised eigenvectors are:

3. A real symmetric matrix $A$ is positive definite if and only if:

4. In a yield curve PCA, the first principal component $q_1$ typically explains approximately what fraction of the variance, and corresponds to which market movement?

5. The variance explained by the first $k$ principal components of a covariance matrix $\hat{\Sigma}$ with eigenvalues $\lambda_1 \geq \cdots \geq \lambda_n$ is:

6. A risk model uses a covariance matrix estimated from $T = 40$ daily returns on $n = 200$ assets. After computing the spectral decomposition, how many eigenvalues are guaranteed to be exactly zero?

7. The best rank-$k$ approximation to a symmetric matrix $A$ in the Frobenius norm is $A_k = \sum_{i=1}^k \lambda_i q_i q_i^\top$. The approximation error $\|A - A_k\|_F$ equals:

8. A portfolio manager asks why two principal components computed yesterday look completely different from the ones computed today, even though the covariance matrix barely changed. The correct explanation is: