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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.

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  • $\begingroup$ 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. $\endgroup$ Oct 6, 2017 at 4:55
  • $\begingroup$ Linear Algebra Done Right - Sheldon Axler $\endgroup$
    – AnlamK
    Oct 6, 2017 at 4:57
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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. $\endgroup$
    – user169852
    Oct 6, 2017 at 6:46
  • $\begingroup$ @MorganRodgers I agree there may some things that are on his list but missing in the book but as far as I could tell a great majority of the listed items are in that book. And I think LADR is one of the most lucidly written linear algebra texts. $\endgroup$
    – AnlamK
    Oct 6, 2017 at 13:42
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    $\begingroup$ Check this $\endgroup$
    – Artem
    Sep 12, 2021 at 22:36

3 Answers 3

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

Reviews by:

EDIT: The second-edition is now titled Matrix Mathematics: A Second Course in Linear Algebra.

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I loved Linear Algebra by Friedberg Insel and Spence

The book by Lax is also excellent.

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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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