Eigenvalues, Eigenvectors and Diagonalization: every key term you need (+ practice quiz)
25 flashcard terms for Linear Algebra Topic 7, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A nonzero vector whose image under a square matrix is a scalar multiple of itself, so the matrix merely stretches or reverses it without changing its direction line.
Eigenvalue
The scalar factor by which a matrix stretches a matching eigenvector. It may be zero, negative or complex even when the matrix has only real entries.
Eigenspace
The set of all vectors satisfying the eigenvector equation for a fixed eigenvalue, together with the zero vector. It is the null space of the matrix minus that eigenvalue times the identity.
Characteristic polynomial
The determinant of the matrix minus a variable times the identity, regarded as a polynomial. Its degree equals the size of the matrix and its roots are the eigenvalues.
Characteristic equation
The equation setting the characteristic polynomial to zero. A scalar is an eigenvalue exactly when it satisfies this equation, since the shifted matrix must be singular.
Algebraic multiplicity
The number of times an eigenvalue occurs as a root of the characteristic polynomial. Counted this way, an n by n matrix over the complex numbers has exactly n eigenvalues.
Geometric multiplicity
The dimension of the eigenspace of an eigenvalue. It is always at least one and never exceeds the algebraic multiplicity of the same eigenvalue.
Defective matrix
A square matrix having some eigenvalue whose geometric multiplicity falls short of its algebraic multiplicity, so no basis of eigenvectors exists.
Independence of eigenvectors
Eigenvectors belonging to distinct eigenvalues are always linearly independent, proved by taking a shortest dependence relation and applying the matrix to it.
Diagonalizable matrix
A square matrix similar to a diagonal matrix, equivalently one whose eigenvectors can be chosen to form a basis of the whole space.
Diagonalization factorization
A writing of a matrix as an invertible matrix of eigenvectors times a diagonal matrix of eigenvalues times the inverse of the eigenvector matrix.
Distinct eigenvalue criterion
A matrix of size n with n different eigenvalues is diagonalizable, since the corresponding eigenvectors are independent and therefore form a basis.
Diagonalizability criterion by multiplicity
A matrix is diagonalizable exactly when the characteristic polynomial factors completely and every eigenvalue has geometric multiplicity equal to its algebraic multiplicity.
Powers of a diagonalizable matrix
Once diagonalized, a high power is computed by raising only the diagonal entries to that power, since the conjugating factors cancel in the middle.
Eigenvalues of a triangular matrix
They are exactly the diagonal entries, because the shifted matrix is again triangular and its determinant is the product of the shifted diagonal.
Trace and determinant from eigenvalues
The sum of the eigenvalues counted with algebraic multiplicity equals the trace, and their product equals the determinant of the matrix.
Zero as an eigenvalue
A matrix has zero as an eigenvalue exactly when it is singular, since a nonzero vector is then annihilated and the null space is the matching eigenspace.
Complex eigenvalues of a real matrix
They occur in conjugate pairs, and the corresponding real transformation acts as a rotation combined with a scaling on a two dimensional invariant plane.
Similarity preserves eigenvalues
Conjugate matrices have the same characteristic polynomial and therefore the same eigenvalues with the same multiplicities, though their eigenvectors differ by the change of basis.
Invariant subspace
A subspace mapped into itself by a transformation. Eigenspaces are the one dimensional examples, and their existence makes block structures possible.
Cayley Hamilton theorem
Every square matrix satisfies its own characteristic equation, so substituting the matrix into its characteristic polynomial yields the zero matrix.
Spectrum of a matrix
The collection of all eigenvalues of a square matrix. Its largest absolute value governs the growth or decay of repeated applications of the matrix.
Dynamical system reading
For a state updated by repeated multiplication, writing the initial state in an eigenvector basis makes each component evolve independently as a geometric sequence.
Eigenvalues under matrix powers
If a scalar is an eigenvalue of a matrix, its kth power is an eigenvalue of the kth power of the matrix with the same eigenvector.
Generalized eigenvector
A vector annihilated by some power of the shifted matrix but not by the first power; such vectors repair the shortage in defective matrices.