Triangle inequality with spectral norm 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. 
 A: Suppose that $A$ is a real $n \times n$ matrix.
Then the $2$-norm or the spectral norm of $A$ is defined as
$$
\Vert A \Vert_2 = + \sqrt{\lambda_{\max}(A^T A)} \tag{1}
$$
where $\lambda_{\max}(A^T A)$ denotes the largest eigenvalue of the non-negative definite matrix $A^T A$ which has real and non-negative eigenvalues.
It is easy to show that
$$
\Vert A \Vert_2 = \sup_{\Vert x \Vert_2 = 1} \ \Vert A x \Vert_2 \tag{2}
$$
so that $\Vert A \Vert_2$ is a subordinate matrix norm.
Proof 1 for Triangle Inequality for $2$-norm:
The triangle inequality for the $2$-norm of matrices can be easily established using (2).
Suppose that $A$ and $B$ are real $n \times n$ matrices.
Then for any vector $x$ satisfying $\Vert x \Vert_2 = 1$, we have
$$
\Vert (A + B) x \Vert_2 = \Vert A x + B x \Vert_2 \leq \Vert A x \Vert_2 + \Vert B x \Vert_2 \tag{3}
$$
From (3), it is immediate that
$$
\sup_{\Vert x \Vert_2 = 1} \ \Vert (A + B) x \Vert_2 \leq
\sup_{\Vert x \Vert_2 = 1} \Vert A x \Vert_2 + \sup_{\Vert x \Vert_2 = 1} 
 \Vert B x \Vert_2 \tag{4}
$$
which shows that
$$
\Vert A + B \Vert_2 \leq \Vert A \Vert_2 + \Vert B \Vert_2
$$
Proof 2 for Triangle Inequality for $2$-norm:
We can rewrite (1) using the SVD of $A$ as
$$
\Vert A \Vert_2 = \sigma_{\max}(A) \tag{5}
$$
where $\sigma_{\max}(A)$ denotes the largest singular value of $A$.
An important identity for the singular values of $A$, $B$, $A+B$ is established in Theorem 3.3.16 (Horn and Johnson, Matrix Analysis):
$$
\sigma_{\max}(A + B) \leq \sigma_{\max}(A) + \sigma_{\max}(B)  \tag{6}
$$
From (6), it is immediate that
$$
\Vert A + B \Vert_2 \leq \Vert A \Vert_2 + \Vert B \Vert_2
$$
which is the triangle inequality for square matrices in $2$-norm.
(The proof given in Horn & Johnson's book also makes uses of the property (2) for the $2$-norm of matrices.)
A: 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.
