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The matrix $M$ below is the matrix of a linear operator $T$ over $\mathbb{R}^4$ with respect to the canonical basis $\{e_1, e_2, e_3, e_4\}$. $$ M = \sqrt{2} \begin{bmatrix} 1/2 & -1/3 & -1/3 & 1/6 \\ 1/3 & 1/2 & -1/6 & -1/3 \\ 1/3 & 1/6 & 1/2 & 1/3 \\ -1/6& 1/3 & -1/3 &1/2 \end{bmatrix}.$$ I proved that $T$ is an isometry showing that the set $\{T(e_1), T(e_2), T(e_3), T(e_4)\}=\{Me_1, Me_2, Me_3, Me_4\}$ is orthonormal. So, by Theorem 9.36 of Axler [1], there is an orthonormal basis of $\mathbb{R}^4$ with respect to which $T$ has a block diagonal matrix such that each block on the diagonal is a 1-by-1 matrix containing $1$ or $-1$ or is a 2-by-2 matrix of the form $$ \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix},$$ with $\theta \in (0, \pi)$. How can I identify this basis?

[1] Axler, S. Linear Algebra Done Right. $3^{rd}$ edition. Springer: New York, 2015.

math_lover0105
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    That is one of the worst Linear algebra textbooks. But find the eigenvalues and eigenvectors. All eigenvals have abs value $1$. Real ones correspond to blocks of size $1$. Complex ones are $cos \alpha +i \sin \alpha$ and correspond to blocks of size $2$. – markvs Nov 09 '21 at 14:51
  • @markvs Now that's a hot take. On this site at least, it seems that the majority opinion is that it's one of the best (see this post for instance) – Ben Grossmann Nov 09 '21 at 18:40
  • @BenGrossmann: I have been teaching the subject for more than 30 years in various Universities, and used many books. This one is the worse. – markvs Nov 09 '21 at 19:06
  • @markvs I've taught the subject a few times myself (once as an instructor, but more often as a TA) and I've tutored for it plenty. If you don't mind expanding, I'd be curious as to what your misgivings are when it comes to LADR. I don't want to occupy the comments here with an off-topic discussion, so I've opened a dedicated chatroom here – Ben Grossmann Nov 09 '21 at 19:17
  • @BenGrossmann: This book is a light version of classical abstract Linear Algebra books, so it is not suitable for a rigorous classical course, and it lacks applications (SVD, for example), so it is bad for any computation-oriented course. Usual books are good at at least one of these. – markvs Nov 09 '21 at 19:23
  • @markvs Thanks for clarifying. For what it's worth, my two cents. I'm not quite sure what a "classical course" would entail in this context; that said, I admit that the book covers no applications and throws out anything regarding fields besides $\Bbb C$. Even so, I think the book is useful preparation for functional analysis because of its emphasis on a "coordinate-free" perspective. I also think that it's brevity makes it meager to build a course around but less intimidating for self-study. I understand why this book might be "the worst" to build a course around, but not "the worst" period – Ben Grossmann Nov 09 '21 at 19:39
  • @markvs But I suppose that could change if I ever had to teach out of it – Ben Grossmann Nov 09 '21 at 19:42
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    One of the comments above may lead readers to believe that Linear Algebra Done Right does not cover SVD. That is not true, however. See the book's Section 7D, which is titled "Polar Decompositions and Singular Value Decomposition". Linear Algebra Done Right has been immensely successful, with multiple textbook adoptions at Berkeley, Harvard, MIT, Princeton, Stanford, and hundreds of other universities (see complete list and excerpts from reviews at https://linear.axler.net/). The book almost always has the best sales rank on Amazon among textbooks for a second course in linear algebra. – Sheldon Axler Nov 10 '21 at 07:53

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