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Eigenvalues
We come up with the terms eigenvalues and eigenvectors when we study the linear transformations. Some vectors change almost by their scale factors when some linear transformation (matrix) is applied to it. Such vectors are known as eigenvectors and the corresponding scale factors are known as eigenvalues of matrix.
Let us learn more about the eigenvalues of matrix along with their properties and examples.
What are Eigenvalues of Matrix?
The eigenvalues of matrix are scalars by which some vectors (eigenvectors) change when the matrix (transformation) is applied to it. In other words, if A is a square matrix of order n x n and v is a nonzero column vector of order n x 1 such that Av = λv (it means that the product of A and v is just a scalar multiple of v), then the scalar (real number) λ is called an eigenvalue of the matrix A that corresponds to the eigenvector v.
The word "eigen" is from the German language and it means "characteristic", "proper", or "own". Thus, eigenvalues are also known as "characteristic values" (or) "characteristic roots" (or) "proper values", etc. The mathematical definition of eigenvalues is as shown below.
How to Find Eigenvalues?
From the definition of eigenvalues, if λ is an eigenvalue of a square matrix A, then
Av = λv
If I is the identity matrix of the same order as A, then we can write the above equation as
Av = λ (Iv) (because v = Iv)
Av  λ (Iv) = 0
Taking v as common factor,
v (A  λI) = 0
This represents a homogeneous system of linear equations and it has a nontrivial solution only when the determinant of the coefficient matrix is 0.
i.e., A  λI = 0
This equation is called the characteristic equation (where A  λI is called the characteristic polynomial) and by solving this for λ, we get the eigenvalues. Here is the stepbystep process used to find the eigenvalues of a square matrix A.
 Take the identity matrix I whose order is the same as A.
 Multiply every element of I by λ to get λI.
 Subtract λI from A to get A  λI.
 Find its determinant.
 Set the determinant to zero and solve for λ.
Let us apply these steps to find the eigenvalues of matrices of different orders.
Eigenvalues of a 2x2 Matrix
Let us see the process of finding the eigenvalues of a 2x2 matrix with an example where we will find the eigenvalues of A = \(\begin{equation}
\left[\begin{array}{ll}
5 & 4 \\ \\
1 & 2
\end{array}\right]
\end{equation}\). Let λ represents its eigenvalue(s). The identity matrix of order 2x2 is, I = \(\begin{equation}
\left[\begin{array}{ll}
1 & 0 \\ \\
0 & 1
\end{array}\right]
\end{equation}\). Then
λI = λ \(\begin{equation}
\left[\begin{array}{ll}
1 & 0 \\ \\
0 & 1
\end{array}\right]
\end{equation}\) = \(\begin{equation}
\left[\begin{array}{ll}
λ & 0 \\ \\
0 & λ
\end{array}\right]
\end{equation}\)
A  λI = \(\begin{equation}
\left[\begin{array}{ll}
5 & 4 \\ \\
1 & 2
\end{array}\right]
\end{equation}\)  \(\begin{equation}
\left[\begin{array}{ll}
λ & 0 \\ \\
0 & λ
\end{array}\right]
\end{equation}\)
= \(\begin{equation}
\left[\begin{array}{ll}
5 λ& 4 \\ \\
1 & 2  λ
\end{array}\right]
\end{equation}\)
Its determinant is,
A  λI = (5  λ) (2  λ)  (1)(4)
= 10  5λ  2λ + λ^{2}  4
= λ^{2}  7λ + 6
The characteristic equation is,
A  λI = 0
λ^{2}  7λ + 6 = 0
(λ  6)(λ  1) = 0
λ  6 = 0; λ  1 = 0
λ = 6; λ = 1
Thus, the eigenvalues of matrix A are 1 and 6.
Eigenvalues of a 3x3 Matrix
Let us just observe the result of A  λI in the previous section. Isn't it just the matrix obtained by subtracting λ from all diagonal elements of A? Yes, so we will use this fact here and find the eigenvalues of 3x3 matrix A = \(\begin{equation}
\left[\begin{array}{lll}
3 & 1 & 1 \\
2 & 4 & 2 \\
1 & 1 & 3
\end{array}\right]
\end{equation}\).
The characteristic equation is,
A  λI = 0
\(\begin{equation}
\left\begin{array}{lll}
3  λ & 1 & 1 \\
2 & 4  λ & 2 \\
1 & 1 & 3  λ
\end{array}\right
\end{equation}\) = 0
(3  λ) [(4  λ)(3  λ)  2(1) ]  1 [ 2(3  λ)  2(1) ] + 1 [2 (1)  1 (4  λ) ] = 0
(3  λ) [12  4λ  3λ + λ^{2}  2]  6 + 2λ + 2 + 2  4 + λ = 0
(3  λ) [10  7λ + λ^{2}]  6 + 3λ = 0
30  21λ + 3λ^{2 } 10λ + 7λ^{2}  λ^{3}  6 + 3λ = 0
λ^{3} + 10λ^{2}^{ } 28λ + 24 = 0
Multiplying both sides by 1,
λ^{3}  10λ^{2}^{ }+ 28λ  24 = 0
This is a cubic equation. We will find one of its roots by trial and error and the other roots by synthetic division. By trial and error, we can see that λ = 2 satisfies the above equation (substitute and check whether we get 0 = 0). Now, using synthetic division,
Set the quotient equal to 0.
λ^{2}  8λ + 12 = 0
(λ  6)(λ  2) = 0
λ = 6; λ = 2
Thus, the eigenvalues of the given 3x3 matrix are 2, 2, and 6.
Properties of Eigenvalues
 A square matrix of order n has at most n eigenvalues.
 An identity matrix has only one eigenvalue which is 1.
 The eigenvalues of triangular matrices and diagonal matrices are nothing but the elements of their principal diagonal.
 The sum of eigenvalues of matrix A is equal to the sum of its diagonal elements.
 The product of eigenvalues of matrix A is equal to its determinant.
 The eigenvalues of hermitian and symmetric matrices are real.
 The eigenvalues of skew hermitian and skewsymmetric matrices are either zeros are purely imaginary numbers.
 A matrix and its transpose have the same eigenvalues.
 If A and B are two square matrices of the same order, then AB and BA have the same eigenvalues.
 The eigenvalues of an orthogonal matrix are 1 and 1.
 If λ is an eigenvalue of A, then kλ is an eigenvalue of kA, where 'k' is a scalar.
 If λ is an eigenvalue of A, then λ^{k} is an eigenvalue of A^{k}.
 If λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^{1} (if the inverse of A exists).
 If λ is an eigenvalue of A, then A / λ is an eigenvalue of the adjoint of A.
Apart from these properties, we have another theorem related to eigenvalues called the "CaleyHamilton Theorem". It says, "every square matrix satisfies its characteristic equation". i.e., if A is a square matrix then it satisfies A  λI = 0. For example, if λ^{2}  8λ + 12 = 0 is the characteristic equation of a square matrix A, then A^{2}  8A + 12 = 0.
Applications of Eigenvalues
 Eigenvalues are used in electric circuits, quantum mechanics, control theory, etc.
 They are used in the design of car stereo systems.
 They are also used to design bridges.
 It is not surprising to know that eigenvalues are also used in determining Google's page rank.
 They are used in geometric transformations.
Examples of Eigenvalues

Example 1: Find the eigenvalues of the matrix \(\begin{equation}
\left[\begin{array}{lll}
3 & 0 & 0 \\
1 & 2 & 0 \\
2 & 0 & 3
\end{array}\right]
\end{equation}\).Solution:
Method 1:
The given matrix is a lower triangular matrix. Hence its eigenvalues are nothing but its diagonal elements which are 3, 2, and 3.
Method 2:
The characteristic equation of the given matrix is:
A  λI = 0
\(\begin{equation}
\left\begin{array}{lll}
3  λ & 0 & 0 \\
1 & 2  λ& 0 \\
2 & 0 & 3  λ
\end{array}\right
\end{equation}\) = 0Expanding the determinant using the first row:
(3  λ)(2  λ)(3  λ) = 0
3  λ = 0; 2  λ = 0; 3  λ = 0
λ = 3; λ = 2; λ = 3
Answer: 3, 2, and 3.

Example 2: Prove that the eigenvalues of the 2x2 matrix \(\begin{equation}
\left[\begin{array}{ll}
0 & 1 \\ \\
1 & 0
\end{array}\right]
\end{equation}\)Solution:
The characteristic equation of the given matrix is:
A  λI = 0
\(\begin{equation}
\left\begin{array}{ll}
0  λ & 1 \\ \\
1 & 0  λ
\end{array}\right
\end{equation}\) = 0\(\begin{equation}
\left\begin{array}{ll}
λ & 1 \\ \\
1 &  λ
\end{array}\right
\end{equation}\) = 0λ^{2} + 1 = 0
λ^{2} = 1
λ = ± √(1) = ± i
But eigenvalues should be real numbers.
Answer: The given matrix has no eigenvalues.

Example 3: Find the eigenvalues of \(\begin{equation}
\left\begin{array}{lll}
1 & 1 & 1 \\
1 & 1 & 1 \\
1 & 1 & 1
\end{array}\right
\end{equation}\).Solution:
The characteristic equation of the given matrix is:
A  λI = 0
\(\begin{equation}
\left[\begin{array}{lll}
1 λ & 1 & 1 \\
1 & 1 λ & 1 \\
1 & 1 & 1 λ
\end{array}\right]
\end{equation}\)(1  λ) [ (1  λ)(1  λ)  1 ]  1 (1  λ  1) + 1 (1  1 + λ) = 0
(1  λ) (1  2λ + λ^{2}  1) + λ + λ = 0
(1  λ) (λ^{2} 2λ) + 2λ = 0
λ^{2}  2λ  λ^{3} + 2λ^{2} + 2λ = 0
λ^{3} + 3λ^{2} = 0
λ^{2} (λ + 3) = 0
λ^{2} = 0; λ + 3 = 0
λ = 0, 0, 3
Answer: 0, 0, 3.
FAQs on Eigenvalues
What is the Definition of Eigenvalues?
Eigenvalues of a matrix are scalars by which eigenvectors change when the matrix or transformation is applied to it. Mathematically, if Av = λv, then
 λ is called the eigenvalue
 v is called the corresponding eigenvector
How can We Find the Eigenvalues of Matrix?
To find the eigenvalues of a square matrix A:
 Find its characteristic equation using A  λI = 0, where I is the identity matrix of same order A.
 Solve it for λ and the solutions would give the eigenvalues.
What are the Eigenvalues of a Diagonal Matrix?
We know that all the elements of a diagonal matrix other than its diagonal elements are zeros. Hence, the eigen values of a diagonal matrix are just its diagonal elements.
How to Find Eigenvalues and Eigenvectors?
For any square matrix A:
 Solve A  λI = 0 for λ to find eigenvalues.
 Solve (A  λI) v = 0 for v to get corresponding eigenvectors.
Where Can We Find Eigenvalue Calculator?
We can find the eigenvalue calculator by clicking here. Here, you can enter any 2x2 matrix, then it will show you the eigenvalues along with steps.
What is Characteristic Equation For Finding Eigenvalues?
If A is a square matrix and λ represents its eigenvalues then A  λI = 0 represents its characteristic equation and by solving this would result in the eigenvalues.
What are the Eigenvalues of an Upper Triangular Matrix?
Since in upper triangular matrix, all elements under the principal diagonal are zeros, the eigenvalues are nothing but the diagonal elements of the matrix.
What are the Eigenvalues of a Unitary Matrix?
A unitary matrix is a complex matrix such that its inverse is equal to its conjugate transpose. The eigenvalues of a unitary matrix are 1 and 1.
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