Eigenvalues (translated from German, this means proper values) are a special set of scalars associated with every square matrix that are sometimes also known as characteristic roots, characteristic values, or proper values. Each eigenvalue is paired with a corresponding set of so-called eigenvectors. The determination of the eigenvalues and eigenvectors of a system is extremely important in physics and engineering, where it arises in such common applications as stability analysis, the physics of rotating bodies, and small oscillations of vibrating systems, to name only a few.

If A is a square \( n \times n \) matrix with real entries and v is an \( n \times 1 \) column vector, then the product w = A v is defined and is another \( n \times 1 \) column vector. It does not matter whether v is real vector v ∈ ℝⁿ or complex v ∈ ℂⁿ. Therefore, any square matrix with real entries (we deal only with real matrices) can be considered as a linear operator A : v ↦ w = A v, acting either in ℝⁿ or ℂⁿ. Of course, one can use any Euclidean space not necessarily ℝⁿ or ℂⁿ.

Although a transformation v ↦ A v may move vectors in a variety of directions, it often happen that we are looking for such vectors on which action of A is just multiplication by a constant. Such a linear transformation is usually referred to as the spectral representation of the operator A. It is important in many applications to determine whether there exist nonzero column vectors v such that the product vector \( {\bf A}\,{\bf v} \) is a constant multiple (which we denote as λ) of v.

Example 1A: Let us consider the following matrix and two vectors:

\[ {\bf A} = \begin{bmatrix} 1 & \phantom{-}2 \\ 4&-1 \end{bmatrix} \qquad \mbox{and} \qquad {\bf v} = \begin{bmatrix} 1 \\ 1 \end{bmatrix} ,\quad {\bf u} = \begin{bmatrix} -1 \\ \phantom{-}1 \end{bmatrix} \]

Then

\[ {\bf A} \, {\bf v} = \begin{bmatrix} 3 \\ 3 \end{bmatrix} = 3 \begin{bmatrix} 1 \\ 1 \end{bmatrix} = 3\,{\bf v} \qquad \mbox{and} \qquad {\bf A} \, {\bf u} = \begin{bmatrix} \phantom{-}1 \\ -5 \end{bmatrix} \]

		line1 = Graphics[{Blue, Thickness[0.01], Line[{{-2, -2}, {2, 2}}]}]; line2 = Graphics[{Blue, Thickness[0.01], Line[{{1, -1}, {-1.3, 1.3}}]}]; line3 = Graphics[{Blue, Thickness[0.01], Line[{{0, 0}, {0.5, -2.5}}]}]; p1 = Graphics[{PointSize[0.03], Point[{{0.7, 0.7}, {2, 2}}, VertexColors -> {Red, Magenta}]}]; p2 = Graphics[{PointSize[0.03], Point[{{-0.7, 0.7}, {0.5, -2.5}}, VertexColors -> {Red, Magenta}]}]; f1 = Interpolation[{{0.7, 0.7}, {0.9, 1.4}, {2, 2}}]; c1 = Plot[f1[x], {x, 0.7, 2}, PlotStyle -> Dashed]; c2 = ListLinePlot[{{-0.7, 0.7}, {-1, -0.2}, {0.5, -2.5}}, InterpolationOrder -> 2, PlotStyle -> Dashed]; t1 = Graphics[{Black, Text[Style["v=(1,1)", Bold, 18], {1.3, 0.7}]}]; t2 = Graphics[{Black, Text[Style["3*v", Bold, 18], {1.6, 2.0}]}]; t3 = Graphics[{Black, Text[Style["u=(-1,1)", Bold, 18], {-1.3, 0.7}]}]; t4 = Graphics[{Black, Text[Style["Au", Bold, 18], {0.1, -2.4}]}]; arx = Graphics[{Arrowheads[0.06], Thick, Arrow[{{-2, 0}, {2, 0}}]}]; ary = Graphics[{Arrowheads[0.06], Thick, Arrow[{{0, -2}, {0, 2}}]}]; Show[line1, line2, line3, p1, p2, c1, c2, t1, t2, t3, t4, arx, ary]
Figure 1: Effects of multiplication by A.		Mathematica code

Example 1B: As another example, we can consider a stochastic matrix that describes the transitions of a Markov chain. Each of its entries is a nonnegative real number representing a probability (in every column, the sum of entries is 1). It is also called a probability matrix, transition matrix, substitution matrix, or Markov matrix.

\[ {\bf A} = \begin{bmatrix} 6/10 & 3/10 \\ 4/10&7/10 \end{bmatrix} = \frac{1}{10} \begin{bmatrix} 6 & 3 \\ 4 &7 \end{bmatrix} . \]

It is easy to verify that matrix A leaves the vector [3, 4] untouched. Indeed,

A = {{6/10, 3/10}, {4/10, 7/10}};
A.{{3}, {4}}

{{3}, {4}}

but shrinks vector [−1, 1]

A.{{-1}, {1}}

{{-(3/10)}, {3/10}}

So we observe that

\[ {\bf A} \begin{bmatrix} 3 \\ 4 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \qquad \mbox{and} \qquad {\bf A} \begin{bmatrix} -1 \\ \phantom{-}1 \end{bmatrix} = \frac{3}{10} \begin{bmatrix} -1 \\ \phantom{-}1 \end{bmatrix} . \]

  

Example 1C: Any constant coefficient difference equation of order n

\[ x_{k+n} = p_{n-1} x_{n+k-1} + p_{n-2} x_{n+k-2} + \cdots + p_{0} x_k , \qquad k=0,1,2,\ldots , \]

where p_i are constants, can be written as a difference equation of first order

\[ {\bf x}_{k+1} = {\bf A}\,{\bf x}_{k} , \qquad k=0,1,2,\ldots , \]

in n dimensional space ℝⁿ, with some n×n matrix A. As a well-known example, we consider the Fibonacci recurrence

\[ F_{k+2} = F_{k+1} + F_k , \qquad F_0 =0, \quad F_1 = 1, \qquad k=0,1,2,\ldots \]

Let L be the linear operator on ℝ² represented by the matrix

\[ L({\bf v}) = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} x+y \\ y \end{bmatrix} , \qquad {\bf A} = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix} , \]

with respect to the standard basis of ℝ². Here v = [x, y]^T is an arbitrary column-vector from ℝ². In particular, for the vector u_k whose coordinates are two consecutive Fibonacci numbers [F_k, F_k-1]^T, we have that

\[ L\left( {\bf u}_k \right) = {\bf A} \begin{bmatrix} F_k \\ F_{k-1} \end{bmatrix} = \begin{pmatrix} 1&1 \\ 1&0 \end{pmatrix} \begin{bmatrix} F_k \\ F_{k-1} \end{bmatrix} = \begin{bmatrix} F_k + F_{k-1} \\ F_k \end{bmatrix} = \begin{bmatrix} F_{k+1} \\ F_{k} \end{bmatrix} = {\bf u}_{k+1} . \]

Thus, we can produce a vector whose coordinates are two consecutive Fibonacci numbers by applying L many times to the vector u₁ with coordinates [F₁, F₀]^T = [1, 0]^T:

\[ L\left( {\bf u}_k \right) = {\bf A}^k \begin{bmatrix} F_1 \\ F_{0} \end{bmatrix} , \qquad k=1,2,\ldots . \]

    ■

There are special classes of matrices for which eigenvalues can be identified uniquely. They are triangular matrices and diagonal matrices. If A is a diagonal, upper triangular, or lower triangular matrix, then entries on its diagonal are its eigenvalues.

Mathematica has some special commands (Eigensystem, Eigenvalues, Eigenvectors, and CharacteristicPolynomial) to deal with eigenvalues and eigenvectors for square matrices. We show how to use them in a sequence of examples. Example 4: 2×2 matrix with complex eigenvalues

The characteristic polynomial can be found either with Mathematica's command CharacteristicPolynomial or multiplying (λ - λ_k)^m for each eigenvalue λ_k of multiplicity m, when eigenvalues are available.Remember that for odd dimensions, Mathematica's command CharacteristicPolynomial provides negative value because it is based on the formula det(A - λI) rather than Eq.\eqref{EqEigen.2}.

======================================================

If A is a square \( n \times n \) matrix and v is an \( n \times 1 \) column vector, then the product \( {\bf A}\,{\bf v} \) is defined and is another \( n \times 1 \) column vector. It is important in many applications to determine whether there exist nonzero column vectors v such that the product vector \( {\bf A}\,{\bf v} \) is a constant multiple (which we denote as λ) of v.

If a homogeneous equation

A = [-2,-4,2;-2,1,2;4,2,5]
[Eigenvalues_A,Eigenvectors_A] = my_eigen(A)
function [eigenvalues, eigenvectors] = my_eigen(A)
%%%This function takes a n by n matrix A and
%%%returns the eigenvalues and their assocaited eigenvectors.
    [m,n] = size(A);
    assert(isequal(m,n), "A is not a square matrix.") %Check to make sure A is a square matrix.
    syms l
    lI = eye(m)*l;
    char_poly = det(A - lI);
    eigenvalues = solve(char_poly,l);
    for i = 1:m %This is to make sure any eigenvalues that = 0 are actually 0.
        if abs(eigenvalues(i)) < 10^-5
            eigenvalues(i) = 0;
        end
    end
    eigenvalues = transpose(eigenvalues); %This makes it easier to know which
                                          %eigenvalue goes with which eigenvector.
    eigenvectors = zeros(m);  %This line is to make the code run smoother.
    for i = 1:m
       matrix = A - eye(m)*eigenvalues(i);
       a_eigenvector = null(matrix); %An eigenvector is the nullspace of a matrix minus one of its eigenvalues along the diagonal.
       a_eigenvector = a_eigenvector/min(abs(a_eigenvector)); %This makes the eigenvector look nice and readable.
       eigenvectors(:,i) = a_eigenvector;
    end
end

Eigensystem[B] gives a list {values, vectors} of the eigenvalues and eigenvectors of the square matrix B.