Why Linear Algebra Done Right didn’t talk about similar matrices

linear algebra

I’m browsing through Axler’s Linear Algebra Done Right, to refresh my linear algebra. It seems adopted a relatively high standpoint view, ie treat matrix as linear mappings , so some proofs are much easier.

However, surprisingly, seems it didn’t talk about similar matrices , i.e. $B=P^{-1}AP$. Why is it not discussed?

Prof Axler definitely knows much more than the book. I’m just curious on the logic behind it,

E.g. if it’s due to there are some equivalent things already discussed in the book that I missed?

or it’s actually not so important or widely used in theory or applications?

or it’s a bit advanced that not in the scope of a book like LADR? In this case , after LADR, which book could be the next one to complet this part ?

Best Answer

Thank you for the question about my book Linear Algebra Done Right. This book devotes major effort to the following question:

  • If $T$ is a linear map from a finite-dimensional vector space $V$ to itself, then what conditions imply the existence of a basis of $V$ with respect to which $T$ has a nice matrix?

Here "nice matrix" might mean, for example, an upper-triangular matrix or a diagonal matrix or a Jordan matrix. Some examples of key theorems in Linear Algebra Done Right that help answer the question above are as follows:

  • If $V$ is a complex vector space, then there exists a basis of $V$ with respect to which $T$ has an upper-triangular matrix.

  • If $V$ is a complex inner product space, then there exists an orthonormal basis of $V$ with respect to which $T$ has an upper-triangular matrix.

  • If $V$ is a real or complex inner product space and $T$ is self-adjoint, then there exists an orthonormal basis of $V$ with respect to which $T$ has a diagonal matrix.

  • If $V$ is a complex inner product space and $T$ is normal, then there exists an orthonormal basis of $V$ with respect to which $T$ has a diagonal matrix.

  • If $V$ is a complex vector space, then there exists a basis of $V$ with respect to which $T$ has a Jordan matrix.

The theorems above all could have been stated in terms of similar matrices, but I preferred to keep the focus on linear maps and bases. The change of basis formula in terms of matrices does appear in Linear Algebra Done Right (see 10.7 in the third edition), although I do not use the "similarity" terminology.

In conclusion, the approach via similar matrices is equivalent to the approach via picking an appropriate basis, but I prefer the basis-centered approach for a second course in linear algebra that focuses on linear maps.

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