TODO

• Block matrix multiplication

• Characteristic polynomial

  • Lemma-1: If \(a_{ij} \in P_1(\lambda)\), then \(|A| \in P_n(\lambda)\)

  • Lemma-2: The characteristic polynomial factors as \((\lambda - a_{11}) \cdots (\lambda - a_{nn}) + p(\lambda)\), where \(p(\lambda) \in P_{n - 2}(\lambda)\)

  • Proofs for \(\text{tr}(A) = \sum \lambda_i\) and \(|A| = \prod \lambda_i\)

• If \(A\) is diagonalizable, then its rank is equal to the number of non-zero eigenvalues. First proof uses the underlying linear map. Second proof uses the equality of multiplicities. If \(A\) is not diagonalizable, the conclusion need not hold. As a counter-example, consider any rotation matrix in \(\mathbb{R}^{2}\).

• Schur’s triangularization

• Spectral theorem

  • Complex version

  • Real version