2. Let $X \in \mathbb{R}^{m \times n}$ be a centered data sample matrix whose covariance matrix is $cov(X) = aI_n + bJ_{n,n}$. Prove that $a \ge 0$ and $b \ge 0$.
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Step 1
First, we need to understand what it means for a matrix to be positive definite. A matrix A is positive definite if for any non-zero vector x, the quadratic form x^T * A * x is positive. In other words, for any non-zero vector x, x^T * A * x > 0. Show more…
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