Optimization_Problem_1

Problem

If $\beta$ is a symmetric matrix, show that there exists a real number $\lambda$ such that $\beta + \lambda I$ is positive definite

Things to consider:

  1. Positive definite means that all eigenvalues of $\beta$ is greater than zero
  2. Thus we want to increase all the eigenvalues of $\beta$ to be greater than zero

Steps:

  1. Get the smallest eigenvalue
$$\lambda_0 < \lambda_1 < \dots < \lambda_N $$


  1. If we apply the absolute value of smallest $\lambda$ value to the equation $\beta + |\lambda| I$ we would get a positive semi definite matrix since $\lambda + |\lambda_1| = 0$, however we want it to be greater than zero


  2. Therefore the number we are looking for is $|\lambda_0| + 1$
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