Use the characteristic equation to show that a and a have th

Use the characteristic equation to show that a and a have the same Eigen-values.

Solution

The matrix (AI)^T is the same as the matrix (A^TI),

since the identity matric is symmetric.

Thus:

det(A^TI) = det((AI)^T) = det(AI)

From this it is obvious that the eigenvalues are the same for both A and A^T

^{IN SIMPLER WAY>>}

I have to show that

det(A-I)=det(A^t-I)

Since,we have

det(A)=det(A^T)

det(A-I)=det((A-I)^T)

det(A^T-I^T)=det(A^T-I)

NOW BY CHARACTERISTIC EQUATIONS

Use the characteristic equation.

Let be an eigenvalue of A.

Then, |A - I| = 0.
Since |M| = |M^T| for any square matrix M,

|(A - I)^T| = 0.

==> |A^_T - ( I)^T| = 0.
==> |A^T - (I)^T| = 0.
==> |A^T - I| = 0, since I = I^T.

Hence is also an eigenvalue for A^T.

Since all the steps above are reversible, we get that any eigenvalue of A^T is also one for A.

So, that\'s how we complete the proof.

Therefore, A and A^T have the same eigenvalues