Anonymous

# Just verifying.?

From the question If a is the eigenvalue of A, then a is also the eigenvalue of AT where T means transpose.

Proof (I've tried answering it not sure if I am right)

If Ax=ax by definition

then Ax-ax=0

=> (A-aI)x=0

=> (A-aI)Tx=0T

=> (AT-aIT)x=0 (since a is just a scalar from the properties of the transpose (aA)T=aAT and IT=I

Thus, ATx=ax. completes the proof!

Relevance

seems correct to me

• Although I don't like to use the notation T for transpose,

I will just stick to the notation (I would use "^t" as in A^t).

Let us look at the following arguments you had:

(A - aI)x = 0 => (A - aI)Tx = 0T.

To go from LHS to RHS, I think that you transpose the

both sides, but the transpose of (A - aI)x is not (A - aI)Tx.

Why? In general, (AB)T is not ATBT. In general, (AB)T

= BTAT.

Now do you see how you can fix the proof?

Let me add few more lines since some people seem confused.

When the questioner deduced (A - aI)Tx = 0T (3rd line from the

deductions) from the second line (A - aI)x = 0, the questioner

was trying to transpose the left hand side and the right hand

side. However, the transpose of (A - aI)x is not (A - aI)Tx as

the questioner wrongfully concluded. Let M = A - aI and N = x

be two matrices, then as I explained above (MN)T = NTMT,

so the traspose of (A - aI)x is xT(A - aI)T, which is a different

matrix from (A - aI)Tx.

Here is a proof of the assertion that the questioner wanted

to prove:

Proof: Let A be a matrix with an eigenvalue a. Then the

determinant of the matrix A - aI is zero, i.e. det(A - aI) = 0.

We know that the determinant is transpose invariant, i.e.

det(C) = det(C^t) (here C^t means the transpose of C).

Thus, det(A - aI) = det((A - aI)^t). On the other hand,

(A - aI)^t = A^t - aI^t = A^t - aI (as two commentators remembered).

Thus, det(A^t - aI) = det(A - aI) = 0. Hence, a is also an

eigenvalue of A^t, the transpose of the matrix A. QED

• Puggy is absolutely correct. The property of transpose on addition or subtraction is distrubitve!

Why that someone would impose the property of transpose on a matrix multiplication beats me unless they were half asleep writing this.

• It looks correct to me as well. I noticed you used the property that

(A + B)^T = A^T + B^T

And this is absolutely correct. What confuses me is how someone claimed you used the property

(AB)^T = (B^T)(A^T)

which you didn't, because you're taking the transpose of two matrices being subtracted (which is a special form of adding).