# Proof: Characteristic polynomial expressed two different ways equals same polynomial.

To my understanding characteristic polynomial can be expressed two different ways and when solving eigenvalues and eigenvectors should give same results. These two ways are:

$$P_A(\lambda)=det(A-\lambda I)$$ $$P_A(\lambda)=det(\lambda I -A)$$

So these two polynomials differ from one another by a sign $(-1)^n$.

We could test this with $2\times2$ matrix A: (because it's easy to compute)

$$A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

$$P_A(\lambda)=(A-\lambda I)$$ $$P_A(\lambda)=det(\begin{bmatrix}a-\lambda & b \\ c & d - \lambda \end{bmatrix})$$ $$P_A(\lambda)= (a-\lambda)(d-\lambda)-bc$$ $$ad + a(-\lambda) + (-\lambda)d + (-\lambda)(-\lambda)-bc$$ $$\lambda^2 -\lambda(a+d) + ad - bc$$

Now we could do the same with second characteristic polynomial:

$$A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ $$P_A(\lambda)=(\lambda I - A)$$ $$P_A(\lambda)=det(\begin{bmatrix} \lambda - a & -b \\ -c & \lambda - d \end{bmatrix}$$ $$P_A(\lambda)=(\lambda-a)(\lambda-d)-(-b)(-c)$$ $$P_A(\lambda)=\lambda\lambda+\lambda(-d)+(-a)\lambda + (-a)(-d)-(-b)(-c)$$ $$P_A(\lambda)=\lambda^2-\lambda(a+d)+ad-bc$$

Now the problem is that this proof applies on 2x2 matrix but how about higher dimensions ? How do you prove this works in higher dimensions as well ? Does such situation exist when these would produce different results ?

If someone could provide some insight on this that would be much appreciated.

Thanks,

Tuki

• You forgot to negate the off-diagonal entries. ;-) It is $\lambda I-A$, so all entries of $A$ have to multiplied by $-1$. – amsmath Oct 19 '17 at 21:18
• To prove it in general, use the fact that $\det(cA)=c^n\det(A)$. – John Griffin Oct 19 '17 at 21:19
• @Tuki: Have you seen Samuelson's Formula for the Characteristic Polynomial: mathworld.wolfram.com/CharacteristicPolynomial.html? Look at the 2x2, 3x3, 4x4, ... – Moo Oct 19 '17 at 21:20
• @Moo no i am not aware of this. Thanks for providing this ! – Tuki Oct 19 '17 at 21:21
• @amsmath i corrected this error on the post and this does not alter the outcome. – Tuki Oct 19 '17 at 21:54