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yousuketakada avatar yousuketakada commented on September 27, 2024

Thank you for your suggestion; your argument totally makes sense.
However, I would suggest a different correction; see my comments below.

I suppose that, throughout the concerned section ``Eigenvector Equation'' beginning at page 698, A is assumed to be square (because eigen things make sense only for squre matrices), and, from the second paragraph at page 699, A is also assumed to be symmetric (although (C.47) and (C.48) hold for non-symmetric matrices as well).
So, it feels more consistent to me to correct as follows:

Page 701
Paragraph 2, Line 1: A matrix $\mathbf{A}$'' should read A symmetric matrix $\mathbf{A}$'' because we usually restrict positive definitness to symmetric matrices only (as assumed in this errata).

As you might have already noticed, a drawback of the above correction is that we should also correct (C.49):

Page 701
Equation (C.49) and the following line: To be consistent with our definition of positive definitness of a square matrix, we should give a symmetric matrix as an example of a non-definite matrix such that all the elements are positive. For instance, a symmetric matrix of the form
\begin{pmatrix}
1 & 2
2 & 1
\end{pmatrix}
has eigenvalues -1 and 3.

It might be helpful to also give an example of a positive-definite matrix such that some (non-diagonal) elements are negative, e.g.,
\begin{pmatrix}
1 & -2
-2 & 5
\end{pmatrix}
has positive eigenvalues.

from prml_errata.

erdosxx avatar erdosxx commented on September 27, 2024

I think that your comment is clear and consistent and can agree with you.
Thank you for you efforts to make PRML more clear. Your errata document is invaluable source to study PRML.

from prml_errata.

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