The eigenvalue is a scalar quantity which is associated with a linear transformation belonging to a vector space. In this article, students will learn how to determine the eigenvalues of a matrix.
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The roots of the linear equation matrix system are known as eigenvalues. It is also considered equivalent to the process of matrix diagonalization.
What are Eigenvalues?
The eigenvalue is explained to be a scalar associated with a linear set of equations which, when multiplied by a nonzero vector, equals to the vector obtained by transformation operating on the vector.
Let us consider k x k square matrix A and v be a vector, then λ is a scalar quantity represented in the following way:
AV = λV
Here, λ is considered to be the eigenvalue of matrix A.
The above equation can also be written as:
(A – λI) = 0
Where “I” is the identity matrix of the same order as A.
This equation can be represented in the determinant of matrix form.
The above relation enables us to calculate eigenvalues λ easily.
Steps to Find Eigenvalues of a Matrix
In order to find the eigenvalues of a matrix, follow the steps below:
Step 1: Make sure the given matrix A is a square matrix. Also, determine the identity matrix I of the same order.
Step 2: Estimate the matrix A – λI, where λ is a scalar quantity.
Step 3: Find the determinant of matrix A – λI and equate it to zero.
Step 4: From the equation thus obtained, calculate all the possible values of λ, which are the required eigenvalues of matrix A.
Decomposition of Eigenvalues
The computation of eigenvalues and eigenvectors for a square matrix is known as eigenvalue decomposition. When we process a square matrix and estimate its eigenvalue equation, and using the estimation of eigenvalues is done, this process is formally termed as eigenvalue decomposition of the matrix.
Eigenvalues so obtained are usually denoted by λ1, λ2, ….. or e1, e2, ….
Properties on Eigenvalues
Let A be a matrix with eigenvalues λ1, λ2, …, λn.
The following are the properties of eigenvalues.
1. The trace of A, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
2. The determinant of A is the product of all its eigenvalues,
3. The eigenvalues of the kth power of A; that is, the eigenvalues of Ak, for any positive integer k, are
4. Matrix A is invertible if and only if every eigenvalue is nonzero.
5. If A is invertible, then the eigenvalues of A-1 are
6. If A is equal to its conjugate transpose, or equivalently if A is Hermitian, then every eigenvalue is real. The same is true of any symmetric real matrix.
7. If A is not only Hermitian but also a positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive or non-negative, negative, or non-positive, respectively.
8. If A is unitary, every eigenvalue has absolute value |λi| = 1.
9. If A is an n × n matrix and {λ1, λ2, …, λk} are its eigenvalues, then the eigenvalues of the matrix I + A (where I is the identity matrix) are {λ1 + 1, λ2 + 1, …, λk + 1}.
Also, read:
Solving linear equations using matrix
Adjoint and inverse of a matrix
Solved Problems on Eigenvalues
Sample problems based on eigenvalue are given below:
Example 1: Find the eigenvalues for the following matrix.
Solution:
Given,
|A – λI| = 0
(2-λ)(5-λ) – 4 = 0
⇒ 10- 5λ – 2λ +λ2-4 = 0
⇒ λ2-7λ +6 = 0
⇒( λ-6)(λ-1) = 0
⇒λ = 6 or λ= 1
Hence the required eigenvalues are 6 and 1.
Example 2: Find the eigenvalues of the matrix
Solution –
Given,
Then,
|A – λI| = 0
(2 – λ) (1 – λ) – 0 = 0
(2 – λ)(1 – λ) = 0
λ = 1, 2
These are required eigenvalues.
Example 3: Calculate the eigenvalue equation and eigenvalues for the following matrix –
Solution –
Let
We can calculate eigenvalues from the following equation:
|A – λI| = 0
(1 – λ) [(- 1 – λ)(- λ) – 0] – 0 + 0 = 0
λ (1 – λ) (1 + λ) = 0
From this equation, we can estimate eigenvalues which are –
λ = 0, 1, -1.
Example 4: Find the eigenvalues for the following matrix.
Solution:
Given,
|A – λI| = 0
(-6 – λ)(5 – λ) – 3 × 4 = 0
⇒ -30 – 5λ + 6λ + λ2 – 12 = 0
⇒ λ2 + λ – 42 = 0
⇒ ( λ – 6)(λ + 7) = 0
⇒ λ = 6 or λ = -7
Hence, the required eigenvalues are 6 and -7.
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Frequently Asked Questions
How do you determine the Eigenvalues of a square matrix A?
We use the equation det(A – λI) = 0 and solve for λ. Calculate all the possible values of λ, which are the required eigenvalues of matrix A.
Give two properties of Eigenvalues.
Matrix A is invertible if and only if every eigenvalue is nonzero.
The trace of A, which is the sum of its diagonal elements, is also the sum of all eigenvalues.
What are the Eigenvalues of a diagonal matrix?
The eigenvalues of a diagonal matrix are its diagonal elements.
Give the characteristic equation to find Eigenvalues.
Let A be a square matrix, and λ represents its eigenvalues, then |A – λI| = 0 represents its characteristic equation.
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