What the Cayley-Hamilton theorem says: every matrix satisfies its own characteristic equation

In the context of linear algebra, how does the Cayley-Hamilton theorem demonstrate the limitation of conceptualizing square matrices solely as linear transformations?

The Cayley-Hamilton theorem shows square matrices as algebraic objects, not just linear transformations

What the rank-nullity theorem says: rank(A) + nullity(A) = n for an m×n matrix

Rank-nullity theorem: Rank(A) + Nullity(A) = Number of columns (n) in A

What the characteristic function φ(t) = E[e^(itX)] does: Fourier transform of the PDF

Characteristic function φ(t) = E[e^(itX)] represents the Fourier transform of the probability density function (PDF)

What CAP theorem states: you can have at most 2 of consistency, availability, partition tolerance

CAP theorem: Consistency, Availability, Partition Tolerance; only 2 can be fully achieved simultaneously

What the Nyquist theorem says: sample at ≥ 2× the highest frequency to avoid aliasing

Nyquist theorem: Sample rate ≥ 2*highest frequency to prevent frequency aliasing

How tiling works in matrix multiplication — loading blocks into shared memory

Tiling in matrix multiplication optimizes cache usage by partitioning matrices into submatrices