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.
The equation for finding eigenvalues of a matrix, is known eigenvalue equation.
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.
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, …., λn, then:
i) Trace of matrix A is equal to sum of its eigenvalues as shown below:
tr (A) = λ1 + λ2 +…. + λn
ii) Determinant of matrix A is equal to product of eigenvalues of A as given below:
det (A) = λ1 . λ2 . …. . λn
iii) Eigenvalues of kthe power of matrix A i.e. Ak 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 eigenvalue equation of given matrix is zero. This happens when there are more than one equilibrium point that lies 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
- AT 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 TA is Rn.
- TA 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.
Dominant and Complex Eigenvalue
Dominant eigenvalue of a matrix is defined to be an eigenvalue which is greatest of all of its eigenvalues.
Let us suppose that A is a square matrix of order n and λ1, λ2, …., λn be its eigenvalues, such that:
λ1 > λ2 >…. > λn
Then, λ1, which is the biggest value of all eigenvalues of matrix A, is known as the dominant 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.
Matrices and Determinants – Important Topics
Matrices and Determinants – Important Questions
Frequently Asked Questions
What do you mean by eigenvalues?
Eigenvalues are the special set of scalars associated with the system of linear equations. It is mostly used in matrix equations.
What do you mean by the dominant eigenvalue of a matrix?
Dominant eigenvalue is defined as an eigenvalue which is greatest of all of its eigenvalues.
Give the eigenvalue equation.
The eigenvalue equation is given by |A-λI |=0. Here I is the identity matrix of the same order as A. λ is the eigenvalue of matrix A.
Can eigenvalue be zero?
Yes, eigenvalue can be zero. But we don’t consider the zero vector to be an eigenvector.