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:

  1. Systems of Linear equations
    1. Diagonalization of a square matrix
    2. Vector Spaces
    3. 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

  1. 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.

  • $\begingroup$ Hoffman and Kunze $\endgroup$ – Bungo Oct 6 '17 at 4:50
  • $\begingroup$ Strang's texts also an option here (at least in terms of topics covered). Not the greatest for proof questions, but he has some interesting conceptual questions, I guess. $\endgroup$ – Omnomnomnom Oct 6 '17 at 4:55
  • $\begingroup$ Linear Algebra Done Right - Sheldon Axler $\endgroup$ – AnlamK Oct 6 '17 at 4:57
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    $\begingroup$ @AnlamK I would argue that that is a particularly bad recommendation for this purpose. My copy is at my office, but I don't think he covers Gaussian elimination, or really most of the things concerning Matrix theory. $\endgroup$ – Morgan Rodgers Oct 6 '17 at 6:17
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    $\begingroup$ Another possibility: take a look at Matrix Analysis and Applied Linear Algebra by Carl Meyer. I believe it covers all of the topics you listed, is quite detailed, and has lots of exercises. And (at least at the time I bought it), it comes bundled with a solution manual, as well as a CD-ROM containing a PDF copy of the book. $\endgroup$ – Bungo Oct 6 '17 at 6:46

I highly recommend A Second Course in Linear Algebra, by Garcia and Horn.

Review by:


The Linear Algebra a Beginning Graduate Student Ought to Know has a pretty good presentation of all this material. It's got some less standard notations and terminology, but it's a good, detailed look at advanced linear algebra and matrix theory.


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