Prove that a diagonal matrix is positive definite if and only if all its diagonal entries are positive. Also write down and identify its associated inner product. (I can't use eigenvalues to prove this because I haven't learned them yet).
In order to be positive definite, matrix K must be symmetric and satisfy positivity. Since we have a diagonal matrix and all its diagonal entries are positive its determinant will be positive as well as its leading coefficient, but how can I show all this information formally using a proof?