6.2 Characteristic Polynomial and Eigenspace

Characteristic polynomials and eigenspaces are key tools for understanding matrices. They help us find eigenvalues and eigenvectors, which reveal a matrix's fundamental properties and behavior.

These concepts are crucial for diagonalization, a process that simplifies matrix operations. By mastering them, you'll unlock powerful techniques for analyzing linear transformations and solving complex problems in linear algebra.

Characteristic polynomial of a matrix

Definition and properties

• The characteristic polynomial of an n × n matrix A is defined as $det(A - λI)$, where $λ$ is a variable and $I$ is the n × n identity matrix
• The degree of the characteristic polynomial is equal to the size of the square matrix A
• The coefficients of the characteristic polynomial are determined by the entries of the matrix A and can be found by expanding the determinant of $A - λI$
• The characteristic polynomial is a monic polynomial, meaning that the leading coefficient (the coefficient of the highest degree term) is always 1
• The constant term of the characteristic polynomial is equal to the determinant of the matrix A

Computing the characteristic polynomial

• To find the characteristic polynomial, replace the main diagonal entries of matrix A with $a_{ii} - λ$, where $a_{ii}$ represents the original diagonal entry
• Expand the determinant of the resulting matrix $A - λI$ to obtain a polynomial in terms of $λ$
• Simplify the polynomial to express it in standard form ($a_nλ^n + a_{n-1}λ^{n-1} + ... + a_1λ + a_0$)
• The coefficients $a_n, a_{n-1}, ..., a_1, a_0$ are determined by the entries of the matrix A
• The characteristic polynomial encodes important information about the matrix A, such as its eigenvalues and their multiplicities

Finding eigenvalues

Characteristic equation

• The characteristic equation is obtained by setting the characteristic polynomial equal to zero: $det(A - λI) = 0$
• Eigenvalues are the roots (solutions) of the characteristic equation
• To find the eigenvalues, solve the characteristic equation for the variable $λ$ using polynomial solving techniques such as factoring, the quadratic formula, or the rational root theorem
• The number of distinct eigenvalues of a matrix is less than or equal to the size of the matrix
• A matrix may have repeated eigenvalues, which occur when the characteristic equation has multiple roots with the same value

Solving the characteristic equation

• Factoring: If the characteristic polynomial can be factored into linear factors, the roots of the equation (eigenvalues) can be found by setting each factor equal to zero and solving for $λ$
• Quadratic formula: If the characteristic polynomial is a quadratic equation ($aλ^2 + bλ + c = 0$), the eigenvalues can be found using the quadratic formula: $λ = \frac{-b ± √(b^2 - 4ac)}{2a}$
• Rational root theorem: If the characteristic polynomial has integer coefficients, the rational root theorem can be used to find potential rational eigenvalues by considering the factors of the constant term and the leading coefficient
• Higher-degree polynomials: For characteristic polynomials of degree 3 or higher, numerical methods or specialized algorithms (cubic formula, quartic formula) may be needed to find the eigenvalues

Algebraic multiplicity of eigenvalues

Definition

• The algebraic multiplicity of an eigenvalue is the number of times the eigenvalue appears as a root of the characteristic equation
• If an eigenvalue is a single root of the characteristic equation, its algebraic multiplicity is 1
• If an eigenvalue is a repeated root of the characteristic equation, its algebraic multiplicity is equal to the number of times it appears as a root
• The sum of the algebraic multiplicities of all eigenvalues is equal to the size of the matrix

Determining algebraic multiplicity

• Factor the characteristic polynomial to identify repeated roots (eigenvalues)
• The exponent of each linear factor $(λ - λ_i)$ in the factored characteristic polynomial represents the algebraic multiplicity of the corresponding eigenvalue $λ_i$
• If the characteristic polynomial cannot be easily factored, the algebraic multiplicity can be determined by finding the roots (eigenvalues) and counting their occurrences
• The algebraic multiplicity provides information about the structure of the matrix and the behavior of the linear transformation it represents

Eigenvectors for eigenvalues

Definition and properties

• An eigenvector of a matrix A corresponding to an eigenvalue $λ$ is a non-zero vector $v$ such that $Av = λv$
• To find eigenvectors associated with an eigenvalue $λ$, solve the equation $(A - λI)v = 0$ for the vector $v$
• The equation $(A - λI)v = 0$ represents a homogeneous system of linear equations, which can be solved using Gaussian elimination or other methods for solving systems of linear equations
• Eigenvectors are not unique; if $v$ is an eigenvector, then any non-zero scalar multiple of $v$ is also an eigenvector corresponding to the same eigenvalue
• Eigenvectors corresponding to distinct eigenvalues are linearly independent

Computing eigenvectors

• Substitute the eigenvalue $λ$ into the equation $(A - λI)v = 0$
• Perform row reduction on the augmented matrix $[A - λI | 0]$ to solve for the vector $v$
• The solution set of the equation $(A - λI)v = 0$ represents the eigenvectors associated with the eigenvalue $λ$
• If the solution set contains free variables, express the eigenvectors in terms of the free variables using parametric vector form
• Normalize the eigenvectors, if desired, by dividing each component by the magnitude (length) of the vector

Eigenspace for an eigenvalue

Definition and properties

• The eigenspace of a matrix A corresponding to an eigenvalue $λ$ is the set of all eigenvectors associated with $λ$, together with the zero vector
• The eigenspace is a subspace of the vector space on which the matrix A operates
• To find the eigenspace, first find the eigenvectors associated with the eigenvalue $λ$ by solving the equation $(A - λI)v = 0$
• The solution set of the equation $(A - λI)v = 0$ forms the eigenspace corresponding to the eigenvalue $λ$
• The dimension of the eigenspace is called the geometric multiplicity of the eigenvalue
• The geometric multiplicity of an eigenvalue is always less than or equal to its algebraic multiplicity

Computing the eigenspace

• Find the eigenvectors associated with the eigenvalue $λ$ by solving the equation $(A - λI)v = 0$
• Express the solution set of the equation $(A - λI)v = 0$ in parametric vector form, using free variables if necessary
• The eigenspace is the span of the linearly independent eigenvectors associated with the eigenvalue $λ$
• To find a basis for the eigenspace, identify the linearly independent eigenvectors from the solution set
• The number of linearly independent eigenvectors in a basis for the eigenspace is equal to the geometric multiplicity of the eigenvalue