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I wanted to write a quick function that calculates the logarithmic matrix norm with respect to the spectral norm. The formula is

$$ \mu_2(A) = \lambda_\mathrm{max}\left(\frac{A + A^T}2\right). $$

So I implemented it by

lognorm[A_] = First@Eigenvalues[(N[A] + Transpose[A])/2, 1];

Unfortunately, for lognorm[{{1, 2}, {3, 4}}] I get {{1, 5/2}, {5/2, 4}} as result. If I use

lognorm2[A_] = Eigenvalues[(N[A] + Transpose[A])/2, 1];

I get for lognorm2[{{1,2},{3,4}}] the result $\{\frac{1}{2} (5+\sqrt{34})\}$, which is strange because it is an exact result, although I added N in the definition of lognorm and lognorm2.

What is the reason for this and how can I write a lognorm function that works properly?

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I found the answer: I have to use deferred definitions with :=. Otherwise Mathematica would calculate

First@Eigenvalues[(N[A] + Transpose[A])/2, 1]

which gives

1/2 (A+Transpose[A])

for some reason and then bind this to lognorm.

If I use := instead, the right hand side is evaluated each time the function is actually called.

See the documentation: http://reference.wolfram.com/language/tutorial/ImmediateAndDelayedDefinitions.html


Also note that Eigenvalues sorts the Eigenvalues by their absolute value, which means that Eigenvalues[{{1, 0}, {0, -5}},1] gives-5 instead of 1, because $|-5|>|1|$. A correct implementation would be

lognorm[A_] := Max[Eigenvalues[(N[A]+Transpose[A])/2]];
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