I am currently working on a question about proving the sum of eigenvalues and I have been searching for the solution from YouTube.
However, I do not understand why the teacher uses the diagonal method to show that the sum of eigenvalues equals the trace of matrix $A$. Doesn't the diagonal method only apply on $3 \times 3$ or less dimensional matrix?
Thank you so much.
Best Answer
If we already know that the determinant of a square matrix is , up to sign, the sum of some products of the matrix's entry, all of which are formed by taking exactly one entry from each row and eactly one entry from each column, then it is not that hard: observe the coefficient of $\;\lambda^{n-1}\;$ in the char. polynomial $\;p_A(\lambda):=| \lambda I - A | \;$ of the matrix $\;A\;$ .
Observe what my definition of the char. pol. is: in the video they use $\; |A- \lambda I|\;$ , which always yields the "problem" of having either $\;1\;$ or $\;-1\;$ as the main coefficient of the pol., and this is avoided with the definition I use.
In order to get the coefficient of $\;\lambda^{n-1}\;$ in that determinant's expansion, we must first observe that we can write the char. pol. as
$$p_A(\lambda)=\prod_{k=1}^n(\lambda-\lambda_k)=(\lambda-\lambda_1)(\lambda-\lambda_2)\cdot\ldots\cdot(\lambda-\lambda_n)$$
where the $\;\lambda_k\;$ are the matrix's eigenvalues (there may well be repeated values. We don't claim all the $\;\lambda_k\,'$ s are different, of course) . Warning: the above expression happens in a field that contains all the eigenvalues of the given matrix, so it may not be the field over which our matrix is defined! If working over the rational or real fields, the above expression is always true when working over the complex $\;\Bbb C\;$ .
Now, in the above expression evaluate the coefficient of $\;\lambda^{n-1}\;$ : you must take $\;n-1\;$ times $\;\lambda\;$ and once some $\;\lambda_i\;$ . At the end you get $\;-(\lambda_1+\lambda_2+\ldots+\lambda_n)\lambda^{n-1}\;$ . Observe the sign: it is always $\;-$Tr.$\,A\;$ with my definition. With the one used in the video it is always Tr.$\,A\,$ . There you go...