simultaneous diagonalization
Diagonalizing one matrix means finding a basis of eigenvectors for it. Simultaneous diagonalization asks for something stronger: a single basis that diagonalizes several matrices at once. In that shared eigenbasis, every matrix in the family becomes diagonal in the same coordinates — a remarkable alignment that not every collection enjoys.
The clean theorem: a family of matrices is simultaneously diagonalizable (by one invertible P, so P^-1 A P is diagonal for every A in the family) if and only if every matrix in the family is individually diagonalizable AND they all commute pairwise, AB = BA. Commuting is the essential extra ingredient; diagonalizability of each is the baseline.
Intuitively, two commuting operators leave each other's eigenspaces invariant, so within each eigenspace of one operator you can further diagonalize the other, refining a common basis. This is the algebraic backbone of why compatible observables in quantum mechanics can be measured together, and why a commuting family of symmetric matrices shares one orthonormal eigenbasis.
Commuting plus individual diagonalizability is exactly equivalent to a shared diagonalizing basis.
Commuting alone is not enough if the matrices are not individually diagonalizable — two commuting defective matrices may share generalized eigenvectors but no diagonalizing basis. Both conditions are genuinely needed.