Linear algerba: trace of square matrix is a linear functional

[skrat]

Lets define trace for each square matrix $A$ its trace as sum of its diagonal elements, so $tr_{n}(A)=\sum_{j=1}^{n}a_{j,j}$. Now proove that trace is a linear functional for all square matrix.

I would be happy to know what has to be true for anything to be a linear functional?

If I understand correctly, linear functional works on a vector but returns a real or complex number. So linear functional is a scalar product. Now what?

 

[CompuChip]

If I understand correctly, linear functional works on a vector but returns a real or complex number.

That is correct. The trace takes a matrix and returns a number. All matrices form a vector space, which you can show by checking the properties of a vector space. Or if you want to cheat a bit, you could write down all the entries of the matrix in one big vector and consider it an element of $\mathbb{R}^{n^2}$.

So linear functional is a scalar product. Now what?

I'm not sure what you mean by "scalar product", are you talking about the inner product $\vec a \cdot \vec b = \sum a_i b_i$?

I would be happy to know what has to be true for anything to be a linear functional?

It has to satisfy that $\operatorname{tr}_n(A + B) = \operatorname{tr}_n(A) + \operatorname{tr}_n(B)$ and $\operatorname{tr}_n(k A) = k \operatorname{tr}_n(A)$ for all n x n matrices A, B and real numbers k. Those are the two properties that make an arbitrary function $V \to \mathbb{R}$ into a linear functional, see e.g. Wikipedia.

 

[skrat]

I'm not sure what you mean by "scalar product", are you talking about the inner product $\vec a \cdot \vec b = \sum a_i b_i$?

Exactly. That is how we defined linear functional.

So, you are trying to say that the following two rows are a complete proof that trace, defined as a sum of diagonal elements of square matrix, is a linear functional:
$tr_{n}(A+B)=\sum_{j=1}^{n}(a_{j,j}+b_{j,j})=\sum_{j=1}^{n}a_{j,j}+\sum_{j=1}^{n}+b_{j,j}=tr_{n}(A)+tr_{n}(B)$
and
$tr_{n}(\lambda A)=\sum_{j=1}^{n}\lambda a_{j,j}=\lambda \sum_{j=1}^{n}a_{j,j}=\lambda tr_{n}(A)$

Does anything change if $\lambda \in \mathbb{C}$

 

[CompuChip]

Exactly. That is how we defined linear functional.

Hmm, the scalar product is only a linear functional if you fix one of the vectors, e.g. for any fixed real vector $\vec a$ the functions
$$f_{\vec a}(\vec v): \mathbb{R}^n \to \mathbb{R}, \vec v \mapsto \vec a \cdot \vec v$$
and
$$g_{\vec a}(\vec v): \mathbb{R}^n \to \mathbb{R}, \vec v \mapsto \vec v \cdot \vec a$$
are both linear functionals (I'll leave it to you to prove it). The inner product $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ is what we call bilinear, where the bi- indicates that it is linear in both arguments.

So, you are trying to say that the following two rows are a complete proof that trace, defined as a sum of diagonal elements of square matrix, is a linear functional:
$tr_{n}(A+B)=\sum_{j=1}^{n}(a_{j,j}+b_{j,j})=\sum_{j=1}^{n}a_{j,j}+\sum_{j=1}^{n}+b_{j,j}=tr_{n}(A)+tr_{n}(B)$
and
$tr_{n}(\lambda A)=\sum_{j=1}^{n}\lambda a_{j,j}=\lambda \sum_{j=1}^{n}a_{j,j}=\lambda tr_{n}(A)$

Yes, that was what I was saying.

Does anything change if $\lambda \in \mathbb{C}$

Not really, except that you have to make everything complex, e.g. since $\lambda A$ will be an n x n matrix with complex entries you have to extend the definition of tr_n those matrices. Note that the example of the scalar product becomes a bit more involved, as $\vec v \cdot \vec w = \sum_i v_i^* w_i$ has an additional complex conjugate compared to the real case.

 

[skrat]

THANK YOU VERY MUCH. One more question:

How do I show that $<A,B>=tr(AB^{*})$ defines scalar product if A,B are both square matrix with complex elements.

Assuming * here means complex conjugation I started like this:

$tr(AB^{*})=\sum_{i=1}^{n}$$(\sum_{j=1}^{n}a_{i,j}$$\cdot {b_{j,i}^{*}})$ but how do i write $<A,B>$?

 

[CompuChip]

A scalar product (or inner product) should satisfy three properties, e.g. one of them is $\langle A, A \rangle \ge 0$ with equality if and only if A = 0. Can you give me the other two properties?

 

[skrat]

I hope I can:

$<A,B>=(<B,A>)^{*}$ and $<\lambda (A+B),C>=\lambda <A,C>+\lambda <B,C>$

So, the same goes for trace: $tr(AA^{*})=0$, $tr(AB^{*})=tr(BA^{*})$ and $tr(\lambda (A+B^{*}))=\lambda tr(A)+\lambda tr(B^{*} )$?