Undergrad What's the geometric interpretation of the trace of a matrix

[Joker93]

Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped.
Thanks!

 

[fresh_42]

In a sense it is the logarithm of the determinant. $\exp : T_0G \rightarrow G$ maps $0$ to $1$ and the trace zero condition in the tangent space $T_0G$ of a linear group $G$ becomes the determinant one condition in the group: $\det e^A = e^{tr A}$

 

[Joker93]

In a sense it is the logarithm of the determinant. $\exp : T_0G \rightarrow G$ maps $0$ to $1$ and the trace zero condition in the tangent space $T_0G$ of a linear group $G$ becomes the determinant one condition in the group: $\det e^A = e^{tr A}$

So, it is just another measure of the volume change of the parallelepiped defined by the columns of the matrix in question?

 

[Strilanc]

The determinant is the product of the eigenvalues of a matrix. The trace is the sum of the eigenvalues of a matrix.

The trace might be somehow related to the perimeter of a parallelepiped (the sum of the lengths of all the edges, probably divided by $2^n$ for a matrix of size $n$). But probably not the one you get by just using the vectors of the matrix, since their total length differs from the sum of the eigenvalues of the matrix. (Also some of the eigenvalues will be deducting from the perimeter since they may be negative.)

 

[fresh_42]

So, it is just another measure of the volume change of the parallelepiped defined by the columns of the matrix in question?

No.
I just wanted to show a possible connection between the two as functions. You might as well simply say the following:
The determinant is the last coefficient of the characteristic polynomial of a matrix with the vectors considered arranged in columns.
The trace is the second coefficient of it. (The first being $1$.)
Or you can say that the determinant is the product of the eigenvalues and the trace the sum of the eigenvalues of a diagonalizable matrix. (Vieta's formulas)

In general I don't know of any geometric interpretation other than the one I've mentioned.
It's similar to $a\cdot b \cdot c$ and $a+b+c$. The first is a volume, and the second? Would you call this "just another measure of the volume change"?

In the post above I pointed out, that there is a similarity to the equation $\log (a\cdot b \cdot c) = \log a + \log b + \log c$ in terms of determinant and trace and that it is not by chance. Algebraically they both play an important role in really many cases. In my opinion it's helpful to keep these relationships in mind rather than to try and find a geometric equivalence for the trace. Both are important properties of mappings in the first place. (Opinion: The volume is a nice to have, but not the main reason we consider the determinant.)