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MATH
[1251]

Theorems from Eigenvalues and eigenvectors

All theorems from Eigenvalues and eigenvectors.

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The eigenvalue criterion

A scalar λ∈F\lambda\in F is an eigenvalue of AA if and only if

pA(λ)=det⁡(A−λI)=0.p_A(\lambda)=\det(A-\lambda I)=0.

For such a λ\lambda, the eigenvectors are the non-zero vectors in ker⁡(A−λI)\ker(A-\lambda I).

This determinant test is how eigenpairs are computed. Diagonalisation then uses the full list of roots to build an eigenbasis.

  1. [1]:Eigenvalue
  2. [2]:Eigenvector
  3. [3]:Eigenspace
  4. [4]:Characteristic polynomial
  5. [5]:Proof
  6. [6]:The matrix with all entries equal to one

Counting eigenvalues

The characteristic polynomial of an n×nn\times n matrix is degree nn. Over C\mathbb C, it has nn roots counted with multiplicity and these roots are the eigenvalues of the matrix.

The roots exist by the fundamental theorem of algebra. Their moduli and real parts later govern growth in stability of systems.

  1. [1]:Characteristic polynomial
  2. [2]:Fundamental theorem of algebra

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The eigenvalue criterionCounting eigenvalues