Triangle inequality with spectral norm

Question

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.

Answer 1

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.)