Recipe

Compute the operator norm of a diagonal or symmetric matrix

Use when: you need $\lVert A\rVert$ for a diagonal matrix, or a symmetric matrix whose eigenvalues you can find

Goal: the exact value of $\lVert A\rVert$ , found directly from the definition rather than from a formula not introduced yet

Taught in Matrix norms and operator norms

Quick recipe

Check whether $A$ is diagonal
Find the eigenvalues of $A$
Take the largest absolute eigenvalue as $\lVert A\rVert$
Confirm the bound is achieved, not merely an upper limit

1
Look at $A$ directly: if every off-diagonal entry is $0$ , the next two steps aren't needed at all.

Example

$A=\begin{bmatrix}3&0\\0&1\end{bmatrix}$ is diagonal, with entries $3$ and $1$ .
- $A$ is diagonal — the operator norm is the largest absolute value among the diagonal entries; skip straight to that answer.
- $A$ is symmetric but not diagonal — continue to Step 2.
Why: on a diagonal matrix, a unit vector's image length is a weighted combination of the diagonal entries, maximized by putting all the weight on the single largest one.
2
Solve the ordinary characteristic equation $\det(A-\lambda I)=0$ , the same technique from earlier chapters.

Example

For $A=\begin{bmatrix}2&2\\2&-1\end{bmatrix}$ , $\det(A-\lambda I)=\lambda^2-\lambda-6=(\lambda-3)(\lambda+2)=0$ , so $\lambda=3$ or $\lambda=-2$ .

Why: only for a symmetric matrix do the eigenvectors form an orthonormal basis, which the next step needs to read $\lVert Ax\rVert$ directly off the eigenvalues.
3
Compare $|\lambda|$ across every eigenvalue found in Step 2, and keep the largest one.

Example

$|3|=3$ and $|-2|=2$ , so the larger is $3$ .
- every eigenvalue is nonnegative — $\lVert A\rVert=\lambda_{\max}$ directly, with no absolute value needed.
- some eigenvalue is negative — compare $|\lambda_{\min}|$ against $\lambda_{\max}$ and keep whichever is larger.
Why: expanding a unit $x$ in $A$ 's eigenbasis makes $\lVert Ax\rVert^2$ a weighted average of the squared eigenvalues, maximized at the eigenvector of largest magnitude, the same reasoning chapter 11's Rayleigh-Ritz theorem used.
4
Evaluate $Ax$ at the (normalized) eigenvector for the eigenvalue found in Step 3, and check $\lVert Ax\rVert$ matches that number exactly.

Example

For $x=(2,1)/\sqrt5$ (the eigenvector for $\lambda=3$ ), $Ax=(6,3)/\sqrt5$ , with length $\sqrt{45}/\sqrt5=3$ , matching Step 3.

Why: this converts Step 3's claimed value into a directly verified one, catching an arithmetic slip in the characteristic equation or the eigenvalue comparison before it stands as the final answer.

Common mistakes

Reporting $\lambda_{\max}$ as $\lVert A\rVert$ without checking whether a more negative eigenvalue has larger absolute value.
Applying the diagonal shortcut to a matrix that only looks diagonal because the off-diagonal entries happen to be small, not exactly zero.
Using a non-symmetric matrix's eigenvalues directly, when this argument only holds because the eigenvectors are orthonormal.
Skipping the Step 4 check and trusting the eigenvalue comparison alone, missing a sign or arithmetic error earlier.