EM 214 Eigenvectors

Eigenvalue formula proof (not needed)

• Ax = λx

• Ax = λ(I_nx)

• Ax = (λI_n)x

• Ax - (λI_n)x = 0

• (A - λI_n)x = 0

• det(A - λI_n) = 0, by properties

Determinant of nxn matrix

“Relevant minor determinants” are determinants with ij row and column removed, where ij is obtained from its coefficient a_ij.

Eigenvalues

• λ is eigenvalue only if det(A - λI) = 0.

• Subtract λ from diagonal of A, calculate determinant, then set calculated λ expression = 0 & solve for λ.

• For 3×3 matrices multiply matrix by -1 to get λ - a, instead of a - λ, and also use cofactor determinant method for easier factorisation.

Eigenvectors

• After eigenvalues are found plug them into (A - λI)x = 0

• Put new coefficient matrix into REF & find values of {x₁,x₂,…,x_n}.

• Parameter variables (r,s,t,u,v) always present (because LD).

• t [v₁, …], t [w₁, …], where v & w would be found vectors.

• Repeat for each eigenvalue. For imaginary eigenvalues calculate both conjugates.

Eigenspace

• After finding eigenvectors write in format;

  • eg. λ = 1: {rx₁ + sx₂ | r,s ∈ R, (r,s) ≠ (0,0)}, where x would be found.

Finding eigenvalues from given potential eigenvectors

• Use Av = λv

• Multiply original matrix (A) by potential eigenvector (v). If new vector is a scalar multiple of potential eigenvector (v) used, then scalar is eigenvalue (λ) & potential eigenvector (v) used is an eigenvector.

Find characteristic equation

• λ expression obtained from det(A - λI) = 0

• Solving for λ not needed.

Eigenvalues of upper/lower triangular matrix

• Eigenvalues are entries on diagonal

Eigenvectors and LI

• Linear combination of eigenvectors is LI if:

  • There are as many unique eigenvectors as eigenvalues.

• {v₁, v₂, …, v_n} for λ₁, λ₂, …, λ_k, where n = k, is LI.

Determinant from characteristic polynomial

• Set λ = 0, then det(A - 0(I_n)) → det(A) = eg. (0)² + 2(0) - 3 = -3

• AKA plug λ = 0 into characteristic polynomial equation, which then is det(A).

Eigenvalues and invertible matrices

• A nxn matrix is invertible if zero is not an eigenvalue of said matrix.

• For an invertible matrix A, λ is an eigenvalue if 1/λ is an eigenvalue of A^-1.

Characteristic equation of 2×2 matrix

• λ² - tr(A)λ + det(A) = 0, where tr(A) is trace of matrix A.

Eigenvalues and symmetric matrices

• If A is symmetric with real entries, then all eigenvalues are real.