Eigenvalues Properties

A matrix is a two-dimensional array of expressions or numbers, which defines a system of linear equations. The roots of this system are termed as eigenvalues. This article helps students to have a clear idea of eigenvalues properties. In linear algebra, we come across an important topic called matrix (plural – matrices).

Eigenvalues are also known as characteristic values or characteristic roots. In branches like Physics and Engineering, the knowledge of eigenvalues and their calculation is extremely important. In this page, we will discuss eigenvalues properties in detail.

Eigenvalue Equation

The equation to find the eigenvalues of a matrix, is known as an eigenvalue equation.

An Eigenvalue equation is shown below –

Where A is a k × k square matrix.

Two parallel lines | | represent the determinant of expression written within it.

λ denotes the eigenvalue of matrix A.

I is the identity matrix of the same order as A.

Eigenvalue Properties

A few important properties of eigenvalues are as follows:

1) A matrix possesses inverse if and only if all of its eigenvalues are nonzero.

2) Let us consider a (m x m) matrix A, whose eigenvalues are λ₁, λ₂, …., λ_n, then:

i) Trace of matrix A is equal to the sum of its eigenvalues as shown below:

tr (A) = λ₁ + λ₂ +…. + λ_n

ii) Determinant of matrix A is equal to product of eigenvalues of A as given below:

det (A) = λ₁ . λ_{2 .} …. . λ_n

iii) Eigenvalues of k^the power of matrix A i.e. A^k will be

iv) If the matrix A is invertible, then its inverse A^-1 does have eigenvalues;

(3) Eigenvalue can be zero

There may be situations that arise such that zero becomes one of the eigenvalues of a matrix. In this case it is obviously implied that any of the solutions of an eigenvalue equation of a given matrix is zero. This happens when there are more than one equilibrium points that lie at origin (0, 0).

(4) If A is an n × n triangular matrix (upper triangular, lower triangular, or diagonal), then the eigenvalues of A are entries of the main diagonal of A.

(5) If μ ≠ 0 complex number, λ is an eigenvalue of matrix A, and x ≠ 0 corresponding eigenvectors, then μx is a corresponding eigenvector.

(6) If A is an n × n matrix, then the following are equivalent.

• A is invertible.
• λ = 0 is not an eigenvalue of A
• If λ is an eigenvalue of matrix invertible A, and x ≠ 0 corresponding eigenvectors, then 1 / λ is an eigenvalue of A^-1, and x is a corresponding eigenvector.
• det(A) ≠ 0.
• Ax = 0 has only the trivial solution.
• Ax = b has exactly one solution for every n × 1 matrix B
•  A^T A is invertible.
• A is diagonalizable.
• A has n linearly independent eigenvectors.
• The reduced row-echelon form of A is In.
• A is expressible as a product of elementary matrices.
• Ax = b is consistent for every n × 1 matrix b.
• The column vectors of A are linearly independent.
• The row vectors of A are linearly independent.
• The column vectors of A span Rⁿ.
• The row vectors of A span Rⁿ.
• The column vectors of A form a basis for Rⁿ.
• The row vectors of A form a basis for Rⁿ.
• A has rank n.
• A has nullity 0.
• The orthogonal complement of the null space of A is Rⁿ.
• The orthogonal complement of the row space of A is 0.
• The range of T_A is Rⁿ.
• T_A is one-to-one.
• λ = 0 is not an eigenvalue of A.

(7) If an n × n matrix A has n distinct eigenvalues, then A is diagonalizable.

(8) If A is a square matrix, then:

• For every eigenvalue of A, the geometric multiplicity is less than or equal to the algebraic multiplicity.
• A is diagonalizable if and if the geometric multiplicity is equal to the algebraic multiplicity for every eigenvalue.

(9) Let A and B are similar matrices. If the similarity transformations performed by the orthogonal or unitary matrix Q, i.e., if applies

(10) If A is Hermitian (symmetric) matrix, then:

• The eigenvalues of A are all real numbers.
• Eigenvectors from different eigenspace are orthogonal.

For example: If we consider a characteristic polynomial λ² – 4λ = 0. Then the eigenvalues are λ = 0 and λ = 4.

Also Read:

• Eigenvectors of a Matrix
• How to Determine the Eigenvalues of a Matrix

Dominant and Complex Eigenvalue

Dominant Eigenvalue

A Dominant eigenvalue of a matrix is defined to be an eigenvalue which is the greatest of all of its eigenvalues.

Let us suppose that A is a square matrix of order n and λ₁, λ₂, …., λ_n be its eigenvalues, such that:

λ₁ > λ₂ >…. > λ_n

Then, λ_1, which is the biggest value of all eigenvalues of matrix A, is known as the dominant eigenvalue.

Complex Eigenvalue

So far, we know that all the values of λ computed by the eigenvalue equation:

|A – λI| = 0 are known as eigenvalues.

When the characteristic equation, thus solved, gives the roots that are complex in nature, the matrix is said to have complex eigenvalues.

In other words, complex eigenvalues of a matrix are the eigenvalues that are of the form:

λ₁ = a + ib and λ₂ = a – ib

Where “a” and “b” are real and imaginary parts, respectively.

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