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Does a 3×3 diagonal matrix commute with anything but another 3×3 diagonal matrix?
A is a 3×3 diagonal matrix with diagonal entries a not equal to b not equal to c. I have to find all matrices that commute.
Does it commute with anything else besides 3×3 diagonal matrices?
Thanks?
I assume that by “a not equal to b not equal to c,” you mean that a, b, and c are all distinct (as opposed to meaning that a ≠ b and b ≠ c, which isn’t the same thing).
If this is what you mean, then no, A won’t commute with any non-diagonal matrix.
—
Suppose we take the matrix
[r s t]
[u v w]
[x y z].
Let us suppose that this matrix commutes with A.
Multiplying by A on the left gives
[a 0 0][r s t] = [ar as at]
[0 b 0][u v w] = [bu bv bw]
[0 0 c][x y z] = [cx cy cz]
On the other hand, multipying the other way gives
[r s t][a 0 0] = [ar bs ct]
[u v w][0 b 0] = [au bv cw]
[x y z][0 0 c] = [ax by cz]
Thus, since the matrices commute, we have
[ar as at] = [ar bs ct]
[bu bv bw] = [au bv cw]
[cx cy cz] = [ax by cz]
which means that as = bs, at = ct, bu = au, bw = cw, cx = ax, and cy = by.
Now, since a ≠ b, then if as = bs, it must be that s = 0. Also, since a ≠ c, then if at = ct, then it must be that t = 0. Similarly, we have that u = 0, w = 0, x = 0, and y = 0.
Thus, our matrix must be of the form.
[r 0 0]
[0 v 0]
[0 0 z].
—
In other words, any matrix which commutes with A must be a diagonal matrix.
