characteristic equation
The characteristic equation is the single equation that produces every eigenvalue at once: det(A - lambda I) = 0. It is the formal statement of when a number lambda lets A - lambda I collapse a direction to zero — that is, when A genuinely has an eigenvector for lambda. Solving it is the textbook route to finding eigenvalues by hand.
The reasoning is short. A v = lambda v with v nonzero means (A - lambda I) v = 0 has a nonzero solution, which happens exactly when A - lambda I is singular, which happens exactly when its determinant vanishes. So det(A - lambda I) = 0 is the precise condition for lambda to be an eigenvalue. Expanding that determinant gives a polynomial in lambda — the characteristic polynomial — whose degree equals the dimension n of the matrix.
Because the equation has degree n, an n-by-n matrix has n eigenvalues counted with multiplicity (over C). The roots can repeat (giving multiplicity) and can be complex even for real matrices. Solving the characteristic equation is fine for small or structured matrices, but for large ones it is numerically unstable; practical software finds eigenvalues by iterative methods instead, never by literally rooting this polynomial.
Setting the determinant of A - lambda I to zero yields a degree-n equation whose roots are the eigenvalues.
Some books write det(lambda I - A) = 0 instead of det(A - lambda I) = 0. The roots are identical; the two polynomials differ only by a factor of (-1)^n, so the choice is purely cosmetic. The det(lambda I - A) form makes the polynomial monic (leading coefficient 1).