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Ma...tri...ces.. (1 Viewer)

leehuan

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I actually gave up on this question because I still couldn't see anything but looking at it again...



I'm not sure if this is allowed, hence why I'm asking... but would I obtain my required result by just changing the order of summation (with the commutative law of multiplication in R)...?
 

InteGrand

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I actually gave up on this question because I still couldn't see anything but looking at it again...





I'm not sure if this is allowed, hence why I'm asking... but would I obtain my required result by just changing the order of summation (with the commutative law of multiplication in R)...?




 
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laters

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If AB=BA (A, B square) does that necessitate that AB=BA=I? As in, if given AB=BA can I deduce that A and B are inverses?
 

InteGrand

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If AB=BA (A, B square) does that necessitate that AB=BA=I? As in, if given AB=BA can I deduce that A and B are inverses?
No. A counterexample is for instance if A is the zero matrix.

However, if we are given that A and B are square matrices with AB = I, then it turns out that yes, this implies that B and A are inverses and BA = I also.
 

leehuan

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We request at least that B can't be the 0 matrix or the identity matrix itself
 

InteGrand

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We request at least that B can't be the 0 matrix or the identity matrix itself
Still no. A classic counterexample would be that if A and B are both diagonal matrices, then AB = BA, but it need not be the identity matrix.

If A and B are both diagonal matrices, then it is easy to verify that AB is the diagonal matrix whose diagonal elements are the corresponding diagonal elements of A multiplied by those of B. Since multiplication of scalars from the field is commutative, we can switch the order of these elements in AB and then see that this is also BA.

 

InteGrand

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Of course another easy type of counterexample is if A = B. Then clearly AB = BA, but it's easy to see that this need not equal I.
 

leehuan

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Still no. A classic counterexample would be that if A and B are both diagonal matrices, then AB = BA, but it need not be the identity matrix.

If A and B are both diagonal matrices, then it is easy to verify that AB is the diagonal matrix whose diagonal elements are the corresponding diagonal elements of A multiplied by those of B. Since multiplication of scalars from the field is commutative, we can switch the order of these elements in AB and then see that this is also BA.

Oh I wasn't intending to imply that they were the only solutions. I knew there were more counterexamples but couldn't figure out what at the time.
 

leehuan

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In general, which pairs of matrices satisfy commutation again? I forgot
 

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