When do 2 matrices commute

The difficulty is that linear algebra is mostly about understanding terms and definitions, and determining which calculation is needed to arrive at the intended answer.

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Nothing very simple can be said that non-tautologically characterises all commuting pairs of matrices. In fact the statement above about the largest commutative subalgebra is false. If you take the set of matrices whose nonzero entries occur only in a block that touches the main diagonal without containing any diagonal positions then this is always a commutative subalgebra.

And then you can still throw in multiples of the identity matrix. See here. The orthogonal matrices don't commute; in fact, there's a subspace of the orthogonals that's non-commutative! But as we know the symmetry group is non-abelian. So just choose two non-commuting permutations and their corresponding matrices clearly don't commute!

Orthogonal matrices are used in geometric operations as rotation matrices and therefore if the rotation axes invariant directions of the two matrices are equal - the matrices spin the same way - their multiplication is commutative. There are lots of "special cases" that commute. The multiplication of two diagonal matrices, for example. Aside: for any two square invertible matrices, A, B, there is something that can be said about AB vs. If the two matrices have Jordan Normal Forms which have the same block structure.

So there is no group of Matrix pairs that commute. Sign up to join this community. The best answers are voted up and rise to the top. In Section 4 , the previous results are directly extended to the case of complex matrices in two very simple ways, namely, either by decomposing the associated algebraic system of complex matrices into two real ones or by manipulating it directly as a complex algebraic system of equations. Basically, the results for the real case are directly extendable by replacing transposes by conjugate transposes.

Finally, further results concerning the commutators of matrices with matrix functions are also discussed in Section 4. The proofs of the main results in Sections 2 , 3 , and 4 are given in corresponding Appendices A , B , and C. It may be pointed out that there is implicit following duality of the main result. Since a necessary and sufficient condition for a set of matrices to commute is formulated and proven, the necessary and sufficient condition for a set of matrices not to commute is just the failure in the above one to hold.

Proposition 2. Note that according to Proposition 2. Then, Proposition 2. The subsequent mathematical result is stronger than Proposition 2. Theorem 2.

The following properties hold. Expressions which calculate the sets of matrices which commute and which do not commute with a given one are obtained in the subsequent result. The following two basic results are concerned with the commutation and noncommutation properties of two matrices.

Proposition 3. Concerning Proposition 3. Results related to sufficient conditions for a set of matrices to pair wise commute are abundant in literature. For instance, diagonal matrices are always pair wise commuting. Any sets of matrices obtained via multiplication by real scalars with any given arbitrary matrix are sets of pair wise commuting matrices.

Any set of matrices obtained by linear combinations of one of the above sets consists also of pair wise commuting matrices.

Any matrix commutes with any of its matrix functions, and so forth. Another useful test obtained from the following result relies on a necessary condition to elucidate if the given set consists of pair wise commuting matrices.

Theorem 3. And when , we may still have , a simple example of which is provided by. This entry contributed by Ronald M. Gantmacher, F. Providence, RI: Amer.

Taussky, O. Monthly 64 , , Aarts, Ronald M.