Characteristic polynomial

The characteristic polynomial of a square matrix A is det(λI − A), a polynomial whose roots are the eigenvalues of A, counted with algebraic multiplicity.

Determinant: A scalar-valued function of a square matrix that encodes invertibility, signed volume scaling, and other algebraic properties. The characteristic polynomial is defined by taking a determinant.
Trace (linear algebra): The sum of the diagonal entries of a square matrix. The coefficient one degree below the leading term is the negative trace.
Spectral theorem: A theorem giving conditions under which a linear operator has an orthonormal basis of eigenvectors. Its eigenvalues, obtained as polynomial roots, determine the diagonal form.
Minimal polynomial (linear algebra): The monic polynomial of least degree that annihilates a given square matrix. It shares the matrix’s eigenvalue roots but can have lower degree.
Eigenvalue: A scalar λ for which a linear transformation sends some nonzero vector to λ times that vector. Its values are exactly the roots of the characteristic polynomial.
Matrix rank: The dimension of the vector space spanned by a matrix’s columns or rows. The multiplicity of zero as a root is tied to the dimension of the null space.
Linear recurrence with constant coefficients: A sequence relation expressing each term as a fixed linear combination of preceding terms. Characteristic roots determine the forms of solutions to recurrence relations.
Characteristic polynomial of a linear operator: The basis-independent polynomial det(λI − T) associated with an endomorphism of a finite-dimensional vector space. It generalizes the matrix definition beyond a chosen coordinate representation.
Polynomial: An expression formed from constants and variables using addition, subtraction, and multiplication with nonnegative integer powers. The characteristic polynomial is a polynomial in λ.
Cayley–Hamilton theorem: The theorem that every square matrix satisfies its own characteristic polynomial. The polynomial gives a matrix identity when λ is replaced by A.

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