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Let $A$ be a real square matrix, I have to prove that $$\left \| A \right \|=\sqrt{\lambda_{\text{max}}(A^*A)}$$ defines a norm. I don't know how to prove the triangle inequality. I have already proved that $\|A\|=\|A\|_2=\sup_{\|x\|_2=1} \|Ax\|_2$, but the exercise is to prove without using it.

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Preliminary answer: This inequality follows from Theorem 3.3.16 in Horn and Johnson (1994) - Topics in Matrix Analysis, and I think they don't use the equality you said in their proof.

Although I don't understand their proof, I am trying to do so since I have this question too (from research). I will update this answer later, once I understand the proof.

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