Just verifying.?

Question

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!

Answer 1

seems correct to me

Answer 2

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?


*** Addendum ***

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

Answer 3

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.

Answer 4

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).

Answer 5

then Ax-ax=0
=> (A-aI)x=0

what is aI in 2nd step?