Let A & B be matrices such that:

A.B = B.A (with det(B) =/= 0)

Show that B can be written under the form:

B = mA + pI where m & p are real numbers and I is the identity matrix.

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- March 29th 2008, 08:01 PMtombrowningtonmatrix multiplication commutativity
Let A & B be matrices such that:

A.B = B.A (with det(B) =/= 0)

Show that B can be written under the form:

B = mA + pI where m & p are real numbers and I is the identity matrix. - March 29th 2008, 09:42 PMmr fantastic
- March 29th 2008, 10:16 PMtombrownington
Sorry Mr. Fantastic, I didn't quite get you.

Can you elaborate a little. - March 29th 2008, 10:18 PMmr fantastic
- March 29th 2008, 10:42 PMtombrownington
- March 29th 2008, 11:10 PMmr fantastic
You need to make the observation that A commutes only with itself (that is, with A) and I. Once this has been noted, it clearly follows that if A commutes with B then B must be a linear combination of the matrices that commute with A, that is, a linear combination of the matrices A and I.

- March 30th 2008, 12:49 AMOpalg
- March 30th 2008, 02:13 AMmr fantastic