Product of Symmetric and Antisymmetric Matrix

In summary, the conversation discusses the trace of the product of a symmetric matrix A and an antisymmetric matrix B. It is shown that the trace of this product is equal to zero due to the properties of symmetric and antisymmetric matrices. The conversation also mentions the use of indices to prove this, but ultimately uses the properties of trace to show that Tr(AB)=0.
  • #1
ognik
643
2
Hi, I want to show that the Trace of the Product of a symetric Matrix (say A) and an antisymetric (B) Matrix is zero.
$So\: (AB)_{ij}=\sum_{k}^{}{a}_{ik}{b}_{kj} $
$and\: Tr(AB)=\sum_{i=j}^{}(AB)_{ij}=\sum_{i}^{}\sum_{k}^{}{a}_{ik}{b}_{ki} $
$because\:A\:is\:symetric, \: {a}_{ik}= {a}_{ki}\:so\:Tr(AB)=\sum_{i}^{}\sum_{k}^{}{a}_{ki}{b}_{ki}$
Here I am stuck - I want to say that because B is antisymetric, it's diagonal entries must be 0, but I am a bit weak with index notation, and especially with double summation signs - can't see how to show $b_{ki}$ is a diagonal element inside this summation ... I think :-)
 
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  • #2
It is not necessary to use indices. Use the facts that $\operatorname{tr}A=\operatorname{tr}A^T$ and $\operatorname{tr}(AB)=\operatorname{tr}(BA)$.
 
  • #3
Thanks Evgeny, I used Tr(ABT) = Tr(ATB)
Tr(ATB)=Tr(AB) and Tr(ABT)=Tr(A(-B))=-Tr(AB)
So Tr(AB)=-Tr(AB), therefore Tr(AB)=0
But if it can be done along the lines I tried with indexes, I'd really like to see that - I am looking for opportunities to practice Indexing :-)
Also I am still unsure what to do when I come across things like $\sum_{}^{}\sum_{}^{}$
 

FAQ: Product of Symmetric and Antisymmetric Matrix

What is a product of symmetric and antisymmetric matrix?

A product of symmetric and antisymmetric matrix is a mathematical operation that involves multiplying a symmetric matrix and an antisymmetric matrix. This results in a new matrix that has both symmetric and antisymmetric properties.

What is the difference between a symmetric and an antisymmetric matrix?

A symmetric matrix is a square matrix that is equal to its transpose, meaning that its elements are symmetric across the main diagonal. On the other hand, an antisymmetric matrix is a square matrix whose elements are equal to the negative of its transpose, resulting in a matrix that is asymmetric across the main diagonal.

What are the properties of a product of symmetric and antisymmetric matrix?

The product of a symmetric and an antisymmetric matrix is always a square matrix with the same number of rows and columns as the original matrices. It also has the properties of both a symmetric and an antisymmetric matrix, meaning it is both symmetric and antisymmetric across the main diagonal.

How is a product of symmetric and antisymmetric matrix calculated?

The product of a symmetric matrix A and an antisymmetric matrix B is calculated by multiplying A and B, then subtracting the product of B and A. This can be represented as AB - BA. The resulting matrix will have both symmetric and antisymmetric properties.

What is the significance of the product of symmetric and antisymmetric matrix?

The product of a symmetric and an antisymmetric matrix is important in mathematics and physics, particularly in the study of quantum mechanics. It can also be used in solving systems of linear equations and in applications such as image processing and signal analysis.

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