I have given the matrix \begin{equation} A = \left(\begin{array}{cc} 4 & 2 \\ 4 & 6 \end{array}\right) \end{equation} And I have to diagonalize it, but I have no idea from where to start. I need help.
2026-04-03 18:43:16.1775241796
How to diagonalize a matrix?
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First find its eigenvalues, which are the roots of its characteristic polynomial: $$ p(x)=x^2-10x+16=(x-2)(x-8). $$ So the eigenvalues are $2$ and $8$.
Then find non-zero vectors $u$ and $v$ such that $$ Au=2u, \quad Av=8v. $$ Then $A=UDU^{-1}$, where $D$ is diagonal with elements of the diagonal $2$ and $8$, and $U$ is the matrix with columns $u$ and $v$.