Syllabus: Bases, dimension. Subspaces. Norms and inner products. Linear operators. Matrix representations. Similarity and unitary similarity. Dual spaces. Transpose and adjoint. Eigenvalues, singular values and norms of operators. Special classes of operators: hermitian, normal, unitary, positive definite, projections. Spectral theorem. Singular value decomposition. Schur triangular form. QR decomposition. Applications. Commuting operators and simultaneous reduction to diagonal and triangular forms. Additional topics to be chosen from the following (suggested) list: Variational principles for eigenvalues and singular values, The Jordan canonical form; nonnegative matrices and the Perron Frobenius theory; applications of singular value decomposition, discrete Fourier transform.
Prerequisites: Calculus, Real Analysis, Linear algebra.
- S. Axler: Linear Algebra Done Right, Second Edition, UTM, Springer, 1997
- M. E. Taylor, Linear Algebra.
- S. R. Garcia and R. Horn: A Second Course in Linear Algebra.