[Math] Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it

eigenvalues-eigenvectorslinear algebramatrices

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to do the things I describe below.

  1. Find the eigenvalues.

  2. Find a unitary matrix $P$ (if there is any) so that the matrix $(P^{-1})AP$ is diagonal.

  3. Find (if there are any) an identity matrix $Q$ and an upper triangular matrix $R$ so that $A=QR$.

Comments (item by item)

  1. I want to know if there is a better way than calculating $\det(A-\lambda I)$.

  2. Well for this I think I have the answer as the matrix A is symmetric that means that it has 4 distinct eigenvectors that are orthogonal with each other also P a matrix composed by using the eigenvectors as columns gives us that $(P^{-1})AP$ = with the diagonal form of A. And P is unitary as if we take the inner product of all the eigenvectors with each other we get 0 since they are orthogonal with each other. Is there a flaw to the way i am thinking?

  3. I tried to solve this using the Gram–Schmidt process I found the first column of Q but then the numbers get too big and gets hard to compute. I have been thinking maybe symmetric matrices have some better way for QR decomposition

Best Answer

I didn't notice the OP was asking for an easier way to unitary diagonalize a matrix. I'll leave this answer here, just because I find this algorithm interesting.

Regarding points 1) and 2), let me introduce an interesting algorithm to diagonalize a real symmetric matrix using only elementary row operations (taken from Schaum's Outline of Theory and Problems of Linear Algebra, by Lipschutz and Lipson).

Congruence Diagonalization of a Symmetric Matrix

And here is a worked example of the above mentioned algorithm:

enter image description here

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