Prove trace of matrix: Tr(AB) = Tr(BA)

[DrMCoeus]

Homework Statement

[/B]
The trace of a matrix is defined to be the sum of its diaganol matrix elements.
1. Show that Tr(ΩΛ) = Tr(ΩΛ)
2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic)

my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U

Homework Equations

$$
Tr(Ω) = \sum_{i=1} Ω_{ii} \\
I = \sum_{k=1}^n | k >< k |
$$

The Attempt at a Solution

$$
Tr(Ω) = \sum_{i=1} Ω_{ii} \\
Tr(ΛΩ) = \sum_{i=1} (ΛΩ)_{ii} \\
(Ω)_{ij} = < i | Ω | j >
$$

(this is saying that when we take the product of the matrices we sum the diagonal entries where the element is in the ith row of the ith column, I also assume the trace is )

1.
$$
= \sum_{i=1} (Λ Ω)_{ii} \\
= \sum_{i=1} < i |ΛΩ| i > \\
= \sum_{i=1} < i |Λ I Ω| i > \\
= \sum_{k=1} \sum_{i=1} < i |Λ| k >< k |Ω| i > \\
= \sum_{k=1} \sum_{i=1} Λ_{ik} Ω_{ki}
$$

Unsure about how to finish as I think I am on the right track but my thinking is a bit cloudy.

To finish proof i need to show as this is where we end up if we reverse the operators initially as we want:

$$
\sum _{k=1} \sum _{i=1} Λ_{ik} Ω_{ki} = \sum _{k=1} \sum _{i=1} Ω_{ki} Λ_{ik}
$$

So correct me where/if I'm wrong.

The way I'm thinking about it is in terms of matrix multiplication. The trace only sums the diagonal elements of the a matrix. Thus when multiplying two matrices/operators the only terms that 'survive' are terms which end up there as the ith row * the ith column. This is denoted by the fact that ' i ' is the 1st term $$Λ_{ik}$$ denotes the ith row time of Λ times the ith column $$Ω_{ki}$$ when we sum through all the values k = 1 to k = n.

Is it possible that we can just swap the two indexes k & i? as they both go to n? Can I just swap their positions as they are are matrix elements and are thus commutative?

For part 2 i can use a similar method to get to:
$$
\sum _{k=1} \sum _{i=1} \sum _{j=1} Λ_{ik} Ω_{kj} θ_{ji}
$$
I notice that the last index of a matrix matches the first index of the next operator. Which may be a clue to why its cyclic but I can't figure out why.

Apologies for the long winded attempt, I just wanted to be clear for anybody attempting to understand my confusion. In my text this proof is in the Unitary Matrix/ Determinant section. These are obviously relevant in later parts to this question but they don't appear to be relevant here.

Any help greatly appreciated

 

[fresh_42]

It might be easier to work with matrices $A_1,A_2,A_3$ and calculate $\operatorname{tr}(A_1A_2)$ and $\operatorname{tr}(A_1A_2A_3)$. Then the result should be symmetric in $(1,2)$ and cyclic in $(1,2,3)$. In any case, you have finite sums here, so changing the order of summation is no issue and the rest comes with the fact, that your matrix entries are hopefully from a commutative and associative domain.

 

[StoneTemplePython]

My advice would be to only spend much time on the first statement, and you ultimately need to find a way to prove this that is intuitively satisfying to you, as traces are ridiculously useful and the cyclic property is vital. I.e. spend your time on proving

$trace\Big(\mathbf {AB}\Big) = trace\Big(\mathbf {BA}\Big)$

Personally, I like to start simple and build, which in this context means rank one matrices, i.e. that

$trace\Big(\mathbf a \tilde{ \mathbf b}^T \Big) = trace\Big(\tilde{ \mathbf b}^T \mathbf a \Big)$

If you look at matrix multiplication in terms of a series of rank one updates vs a bunch of dot products, you can build on this simple vector setup.

- - - -
once you have this locked down, use that fact plus associativity of matrix multiplication to get the cyclic property. Specifically consider

$trace\Big(\mathbf {XYZ}\Big) = trace\Big(\big(\mathbf {XY}\big)\big(\mathbf Z \big)\Big)$

now call on the fact that $trace\Big(\mathbf {AB}\Big) = trace\Big(\mathbf {BA}\Big)$, which if you assigned $\mathbf A := \big(\mathbf {XY}\big)$ and $\mathbf B := \big(\mathbf Z\big)$, gives you the below:

$trace\Big(\big(\mathbf {XY}\big)\big(\mathbf Z \big)\Big) = trace\Big(\big(\mathbf Z \big)\big(\mathbf {XY}\big)\Big)$

and use associativity and original proof once more to finish this off.

 

[DrMCoeus]

Thanks for help guys. I am beginning to understand some of my issues.

However I realized that that this identity applies for non-square matrix products also. i.e. if we have matrix A (2x3 matrix) and matrix B (3x2 matrix) then AB produces a 2x2 matrix & BA produces a 3x3 matrix yet the traces are still the same. Which was something I hadn't even considered and very interesting.

Personally, I like to start simple and build, which in this context means rank one matrices, i.e. that

$trace\Big(\mathbf a \tilde{ \mathbf b}^T \Big) = trace\Big(\tilde{ \mathbf b}^T \mathbf a \Big)$

If you look at matrix multiplication in terms of a series of rank one updates vs a bunch of dot products, you can build on this simple vector setup.

I was just wondering if you could elaborate on this a bit more? More specifically, why do you have the complex conjugate and transpose of b denoted. Also the term rank one updates is unfamiliar to me. My Linear algebra is geared towards QM and lacks some generality.

Also I had figured out the answer to why they are cyclic. Very nice proof indeed :)

 

[StoneTemplePython]

Thanks for help guys. I am beginning to understand some of my issues.

However I realized that that this identity applies for non-square matrix products also. i.e. if we have matrix A (2x3 matrix) and matrix B (3x2 matrix) then AB produces a 2x2 matrix & BA produces a 3x3 matrix yet the traces are still the same. Which was something I hadn't even considered and very interesting.
I was just wondering if you could elaborate on this a bit more? More specifically, why do you have the complex conjugate and transpose of b denoted. Also the term rank one updates is unfamiliar to me. My Linear algebra is geared towards QM and lacks some generality.

Also I had figured out the answer to why they are cyclic. Very nice proof indeed :)

I would probably start off by assuming the scalars are in $\mathbb R$ for starters here that way you don't get lost in things like dot product versus inner product. If you right click the LaTeX, and do 'show math as -> Tex commands' you'll see that the symbol above the b vector in

$trace\Big(\mathbf a \tilde{ \mathbf b}^T \Big) = trace\Big(\tilde{ \mathbf b}^T \mathbf a \Big)$

is actually the "tilde" sign.

People don't always use it, but basically $\tilde{ \mathbf b}^T$ indicates that the vector is a row vector and was originally a row vector. (As opposed to if it didn't have the tilde sign, it would mean you have a column vector that has been transposed so that it is now a row vector.)

Notation is not uniform though... FWIW if I was showing a conjugate transpose I would do $\mathbf b^H$ or $\mathbf b^*$ but again, in the example I show here, conjugation has nothing to do with anything... you're certainly welcome to ignore the tilde signs if you want. There is a hint, though, tied in with the tilde sign -- consider how to partition $\mathbf A$ and $\mathbf B$ appropriately into row and column vectors, and look at the two different types of multiplication mentioned.

As for rank one updates -- you can just call it a finite series of rank one matrices if you want. Sometimes people call it the outer product interpretation of matrix multiplication.

Again, the key thing is finding a way to prove

$trace\Big(\mathbf {AB}\Big) = trace\Big(\mathbf {BA}\Big)$

in a way that is satisfying and intuitive to you. Most proofs I've seen involve juggling double sigmas. I prefer something a bit more visual, and building off of rank one matrices.

In Linear Algebra Done Wrong, they strongly hint at using, in effect, indicator variables for $\mathbf B$ to prove this, but ultimately the problem is left as an exercise. It is a very good book, and free. You may want to read it as some point. It is available here:

https://www.math.brown.edu/~treil/papers/LADW/book.pdf

 

[fresh_42]

I was just wondering if you could elaborate on this a bit more?

You have already done all the work needed in your opening post under point $1$. Just use $\Lambda_{ik}\Omega_{ki}=\Omega_{ki}\Lambda_{ik}$, switch the summation order and walk the same way back to end up with $\operatorname{Tr}(\Omega \Lambda)$. The $3-$cycles follow with what @StoneTemplePython said in post #3 by using the associativity of matrix multiplication. No need for ranks, row- or column vectors, transposings, conjugates or special fields, except that the scalar field has to be commutative and associative.