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  "body": "\n## Proof\n\n1. Let $A$ be a [[positive definite|0QQCG3JWJ4]] Hermitian $n \\times n$ matrix, and let $\\lambda$ be an [[eigenvalue|0ZND7NQ970]] of $A$.\n2. By definition there is $x \\in \\C^n$ with $x \\ne 0$ and $Ax = \\lambda x$.\n3. Let $x^*$ be the [[conjugate transpose|0EX045E2BP]] of $x$. Then $x^* x = \\sum_{i=1}^n \\abs{x_i}^2$, which is a real number.\n4. Since $x \\ne 0$, some $x_i \\ne 0$, so $x^* x \u003e 0$.\n5. By step 2, $x^* A x = x^* (\\lambda x) = \\lambda\\, x^* x$.\n6. Since $A$ is positive definite and $x \\ne 0$, the number $x^* A x$ is real and positive.\n7. By steps 4 to 6, $\\lambda = x^* A x / x^* x$ is a quotient of two positive reals. So $\\lambda$ is a positive real number.\n\n## Idea\n\nTest the defining inequality $x^* A x \u003e 0$ on an eigenvector. There $A$ acts as multiplication by $\\lambda$, so the inequality becomes $\\lambda \\abs{x}^2 \u003e 0$.\n",
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