Which relationship between trace and determinant is established using density?

Question

I read in some lecture notes that "as an example for the intersection between linear algebra and calculus, one can establish the relationship between trace and determinant of a matrix using a density-argument".

Which relationship is meant? And what would the argument be?

Perhaps they're referring to the formula $$ \det\exp(A) = \exp(\operatorname{trace}(A)) $$ — Ben Grossmann, Oct 03 '18 at 22:49
Another notable relationship is $$ \det ( I_n + hA) = 1 + h , \mbox{tr} (A) + O (h^2) $$ some other suggestions can be taken from this post — Ben Grossmann, Oct 03 '18 at 22:55
@Omnomnomnom Well, are these really known for being verifiable using density? Thank you for the link, it was helpful. — Ramen, Oct 03 '18 at 22:57
the first one certainly is. Not sure about the second off the top of my head — Ben Grossmann, Oct 03 '18 at 22:57
Yes, I now think they must have meant the first one... Thank you @Omnomnomnom — Ramen, Oct 03 '18 at 23:01

score 2 · Accepted Answer · answered Oct 03 '18 at 22:45

2

For example, it can be seen that the set of diagonalizable matrices is dense in the set of $n \times n$ matrices over $\mathbb{C}$. So, when you have a continuous operation on $n \times n$ matrices, you can look at what it does to diagonal matrices and then argue that any other matrix is arbitrarily close to a diagonal matrix in some basis and then use limits to prove the result. For example, $\mathrm{det}$ is continuous on $n \times n$ matrices because the formula is given by a multivariate polynomial. Can you proceed from here?

answered Oct 03 '18 at 22:45

stressed out

8,130

1

I think that in combination with the first comment of @Omnomnomnom this answered my question. Thanks! – Ramen Oct 03 '18 at 23:01
@Ramen You're welcome. – stressed out Oct 03 '18 at 23:03

Which relationship between trace and determinant is established using density?

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