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I really like "Finite-Dimensional Vector Spaces" by Paul Halmos (Undergraduate Texts in Mathematics).

Is there any graduate level version of this book (not necessarily by Halmos), i.e. similar subject and style but at a more advanced level?

I am not really interested in the analysis part (e.g. convergence, completeness etc) but rather more interested in the algebraic parts like Eigenvalues, Eigenvectors and Spectral Theorems.

Thanks for any recommendations.

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    $\begingroup$ By the way, please see the edit in my answer. $\endgroup$ – onurcanbektas Mar 2 '18 at 6:26
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Besides Greub's text, here are a few others that appear to fit your objectives . . .

  • Roman -- Advanced Linear Algebra, 3rd Ed (2008)
      From the preface:

      This book is a thorough introduction to linear algebra, for the graduate or advanced undergraduate student.$\\[10pt]$

  • Blyth & Robertson -- Further Linear Algebra (2001)
      From the preface:

      Our title "Further Linear Algebra" suggests already that the reader will be familiar with the basics of this discipline.$\\[10pt]$

  • Cooperstein -- Advanced Linear Algebra, 2nd Ed (2015)
      From the preface:

      . . . through the choice of various subsets of the chapters, this book can be appropriate for a single upper-division or graduate course in linear algebra, . . .

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I would advise you to try out Greub's Linear Algebra book. It is a graduate text, and the author is very rigorous in the threatment of the subject.

If you like, you can read the short review that I have written about the book from here.

Edit:

In the last year, I have covered more than half of the book, and asked lots of questions in this site about the material, so if you stuck at some point as I have done, you can check out the posts that I have asked, or directly post a question in here (obviously), I would be happy to answer if I can.

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