Powers and repeated transformations

Applying the same map $k$ times corresponds to $A^k$ when well-defined. Diagonalization later makes powers cheap; shear shows nontrivial Jordan behavior .

If $A$ is diagonal with entries $\lambda_i$, then $A^k$ is diagonal with entries $\lambda_i^k$. $(A^2)\mathbf{x}$ means apply $A$ twice. For non-square $A$, $A^2$ need not exist unless inner dimensions match after the first multiply.

Floating point: repeated multiplication squares condition numbers. Watch stability in applications. $(AB)^2$ is generally different from $A^2B^2$ because $ABAB$ differs from $AABB$ unless middle factors commute.

Exponent counts iteration: $A^k\mathbf{v}$ applies the same linear pipeline $k$ times to $\mathbf{v}$ .

Check your understanding. The tasks below rest on these ideas: Correct: powers of a diagonal matrix raise each diagonal entry to that power. Not quite: it is always defined, is not the identity in general, and uses powers, not multiples, of the entries. Correct: $A^2\mathbf{x}=A(A\mathbf{x})$, the same map applied twice. Not quite: it involves no transpose, no averaging, and squaring entries is a different (entrywise) operation. Correct: $A^2$ requires $A$'s shapes to be compatible with itself, which fails unless $A$ is square. Not quite: it does not always exist, symmetry is undefined for non-square $A$, and $A^\top A$ is a different product. Correct: expanding shows the middle factors must be swapped, which needs $BA=AB$. Not quite: squaring products is well defined, the shapes match, and the determinant sign is irrelevant.