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Matrix multiplication: Communicative property.

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m26k9
#1
Jan23-09, 12:25 AM
P: 9
Hello,

First time poster.
I have got a question about commutative property of matrix multiplication.
Literature says that matrix multiplication is communicative only when the two matrices are diagonal.

But, I have a situation with an 'Unitary' matrix. Actually it is the DFT matrix http://en.wikipedia.org/wiki/Discret...he_unitary_DFT. And I multiply with a 'vector'.

It seems that communicative property holds in this case. But I want to know what is the theoretical explanation, or the property as to why communicative property holds in this case.

Thank you very much.
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wsalem
#2
Jan23-09, 04:20 AM
P: 56
Literature says that matrix multiplication is communicative only when the two matrices are diagonal.
You certainly misread that, because it is not true that the diagonal matrix is the [b]only[/tex] type of matrices that are commutative, surely many matrices are not commutative, but some are.
Take for example the set of 2x2 matrices of the form
[tex]\begin{array}{cc}
a & b \\
-b & a
\end{array}
[/tex]
Tac-Tics
#3
Jan23-09, 08:44 AM
P: 810
Another silly counter-example:

Matrices of the form:


[tex]\begin{array}{cc}
a & 0 \\
0 & 0
\end{array}[/tex]

hokie1
#4
Jan23-09, 08:53 AM
P: 42
Matrix multiplication: Communicative property.

The silly counter-example is a diagonal matrix in which one of the entries on the diagonal happens to be 0.

A clarification might be useful here. You have a matrix that is multiplied by a vector? That might be an (m x m) matrix multiplied by an (m x 1) vector. You can't multiply an (m x 1) with an (m x m).
Tac-Tics
#5
Jan23-09, 09:14 AM
P: 810
Quote Quote by hokie1 View Post
The silly counter-example is a diagonal matrix in which one of the entries on the diagonal happens to be 0.

A clarification might be useful here. You have a matrix that is multiplied by a vector? That might be an (m x m) matrix multiplied by an (m x 1) vector. You can't multiply an (m x 1) with an (m x m).
Heh, oops. The silly example is for some reason I was using an alternative definition for "diagonal" X-D
hokie1
#6
Jan23-09, 09:25 AM
P: 42
Thats OK. Usually I'm the one saying oops.
m26k9
#7
Jan23-09, 11:45 AM
P: 9
Thanks a lot for the replies guys.

I am multiplying a (1xM) vector with the (MxM) unitary matrix.
For the few random matrices I tried, it seems to be commutative. Atleast for the DFT matrix (Vandermond) I tried.
I want to know if there is any property talking about this scenario.

Thank you very much.
Peeter
#8
Jan23-09, 02:07 PM
P: 295
Quote Quote by m26k9 View Post
For the few random matrices I tried, it seems to be commutative.
Can your matrixes be diagonalized with the same similarity transformation?

[tex]
\begin{align*}
U_1 &= V D_1 V^* \\
U_2 &= V D_2 V^*
\end{align*}
[/tex]

EDIT: fixed wrong lingo in question.
D H
#9
Jan23-09, 02:58 PM
Mentor
P: 15,167
Quote Quote by m26k9 View Post
Thanks a lot for the replies guys.

I am multiplying a (1xM) vector with the (MxM) unitary matrix.
For the few random matrices I tried, it seems to be commutative.
You are using the term "commutative" incorrectly here. Given an operator * and operands a and b, a and b commute if a*b=b*a. If a is 1xM and b is MxM, the product a*b exists (and is 1xM) but b*a exists only if M=1. In other words, there is no way a 1xM and a MxM matrix can commute unless M=1 (i.e., if a and b are scalars).
hokie1
#10
Jan23-09, 08:59 PM
P: 42
That's what I was getting at. That's why I was mentioning the dimensions of the matrices. Too often software packages bend the rules and treat vectors as both (1 x m) and (m x 1) matrices to fit the math. In that case a symmetric (m x m) matrix allows the operation to be commutative, i.e. A[i,j] = A[j,i].
m26k9
#11
Jan24-09, 02:18 AM
P: 9
Sorry guys.

Yes, if I am using say A is a (1xM) vector, it cannot be commutative.
My mistake. Actually I am AxB and BxA', conjugate of A.
So this is not commutation anymore?
If this is the case, that means matrix multiplication properties cannot be applied, unless I put my vector entries inside a diagonal of a matrix?

Thanks lot guys.


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