I am looking for a book to learn Advanced Linear Algebra and Matrix Theory in detail.
Sheldon Axler :Doesn't cover matrix theory,Hoffman,Kunze:Doesn't have many exercises and examples on each of the topics
Please suggest some alternatives
Requisites: Theorems with proofs,easy ones left to reader,Enough examples,Good Exercises(with Hints if possible)
Topics to cover:
- Systems of Linear equations
- Diagonalization of a square matrix
- Vector Spaces
- Solutions of Linear Systems: Gaussian elimination , Null Space and Range , Rank and nullity, Consistency conditions in terms of rank , General Solution of a linear system , Elementary Row and Column operations , Row Reduced Form ,Triangular Matrix Factorization
5.Important Subspaces associsted with a matrix: Range and Null space, Rank and Nullity,Rank Nullity theorem .
6.Orthogonality: Inner product, Inner product Spaces , Cauchy – Schwarz inequality , Norm , Orthogonality , Gram – Schmidt orthonormalization , Orthonormal basis , Expansion in terms of orthonormal basis – Fourier series , Orthogonal complement.
7.Eigenvalues and Eigenvectors
- Hermitian Matrices:Real symmetric and Hermitian Matrices Properties of eigenvalues and eigenvectors.
9.General Matrices: The matrices $AA^T,A^TA$ Rank, Nullity, Range and Null Space of $AA^T,A^TA$ ,Singular Value Decomposition.
10.Jordan Cnonical form: Primary Decomposition Theorem Nilpotent matrices Canonical form for a nilpotent matrix
Mostly results on MSE said to follow Matrix Analysis-Horn,Johnson but the book does not cover all the topics in great detail.It focuses on more advanced topics.
Please suggest a book accordingly as I need to prepare for my exam.