What is the difference between an eigenvalue and an eigenvector?

An eigenvalue is a scalar that scales the eigenvector when multiplied by the matrix. The eigenvector is the non-zero vector that does not change direction under the transformation.

Can eigenvalues be complex numbers?

Yes. If the matrix has complex entries or if the characteristic polynomial has no real roots, the eigenvalues may be complex.

Are eigenvalues always real for symmetric matrices?

Yes, if a matrix is symmetric (i.e., A^(T) = A ), all its eigenvalues are real.

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Eigenvalues of a Matrix

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Eigenvalues of a Matrix

In linear algebra, eigenvalues are special numbers associated with a square matrix. They reveal important characteristics like stability, direction, and scaling of vector transformations. Widely used in mathematics, physics, and data science, eigenvalues simplify complex matrix operations. Understanding eigen values is key for solving systems of equations, quantum mechanics, and even in machine learning techniques like PCA (Principal Component Analysis).

1.0Definition of Eigenvalues

An eigenvalue of a square matrix A is a scalar λ such that:

$A x = λ x$

Here, A is an n × n matrix, x is a non-zero vector (called the eigenvector), and λ is the eigenvalue. This equation means that multiplying a matrix A by a vector x results in a scaled version of x, not a rotated or otherwise transformed one.

2.0Eigenvalue Formula

To find the eigenvalues of a matrix, use the characteristic equation:

$d e t (A - λ l) = 0$

Where:

A is a square matrix,
I is the identity matrix of the same size,
λ represents eigenvalues,
det stands for determinant.

This results in a polynomial equation in λ. The roots of this polynomial are the eigenvalues of the matrix.

3.0Eigenvalue of a 2×2 Matrix

For a 2×2 matrix: $A = [a c b d]$

The characteristic equation becomes:

$det ([a - λb c d - λ]) = 0$

$\Rightarrow (a - λ) (d - λ) - b c = 0$

$\Rightarrow λ^{2} - (a + d) λ + (a d - b c) = 0$

The roots of this quadratic equation give the eigenvalues of the 2×2 matrix.

4.0Properties of Eigen Values

Here are some key properties of eigen values for square matrices:

Only Square Matrices Have Eigenvalues: Non-square matrices do not have eigenvalues.
Sum of Eigenvalues = Trace of Matrix: $\sum λ_{1} = T r (A)$
Product of Eigenvalues = Determinant of Matrix: $\prod λ_{i} = det (A)$
Eigenvalues of a Diagonal Matrix are the Diagonal Elements.
If A is symmetric, all eigenvalues are real.
Eigenvalues of A⁻¹ are reciprocal of A’s eigenvalues.
If A is triangular, eigenvalues are its diagonal entries.

5.0Example: Finding Eigenvalues of a 2×2 Matrix

Example 1: Let $A = [4123]$

Solution:

Step 1: Set up the characteristic equation:

$det (A - λ I) = 0$

$= det ([4 - λ 1 2 3 - λ])$

$= (4 - λ) (3 - λ) - (2) (1)$

$= λ^{2} - 7 λ + 10 = 0$

Step 2: Solve the quadratic equation:

$λ = \frac{7 \pm 49 - 40}{2} = \frac{7 \pm 3}{2}$

$\Rightarrow λ_{1} = 5, λ_{2} = 2$

Hence, the eigenvalues are 5 and 2.

Example 2: Let $A = 2 - 1 0 - 1 2 - 1 0 - 1 2 .$ Find the eigenvalues of matrix A.

Solution:

This is a symmetric tridiagonal matrix. Use the characteristic equation:

$det (A - λ I) = 0$

$\Rightarrow 2 - λ - 1 0 - 1 2 - λ - 1 0 - 1 2 - λ = 0$

Let’s evaluate the determinant using cofactor expansion:

Let $D (λ) = det (A - λ I)$

Use shortcut tridiagonal determinant formula or compute directly:

$D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (- 1)^{2} (2 - λ) D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (2 - λ) D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (2 - λ)$

Factor out : $(2 - λ)$

$= (2 - λ) [(2 - λ)^{2} - 1 - 1] = (2 - λ) [(2 - λ)^{2} - 2]$

Let $x = 2 - λ$

$\Rightarrow D = x (x^{2} - 2) = x^{3} - 2 x = 0 \Rightarrow x (x^{2} - 2) = 0 \Rightarrow x = 0, x = \pm 2$

So,

$2 - λ = 0 \Rightarrow λ = 22 - λ = 2 \Rightarrow λ = 2 - 22 - λ = - 2 \Rightarrow λ = 2 + 2$

Eigenvalues:

$2, 2 + 2, 2 - 2$

Example 3: Find the eigenvalues of $A = 700 0 - 3 0 005 .$

Solution:

For a diagonal matrix, the eigenvalues are the diagonal elements.

So, the eigenvalues are:

7, −3, 5

Example 4: Let $A = [01 - 1 0] .$ Find the eigenvalues of A.

Solution:

$det (A - λ I) = - λ 1 - 1 - λ = λ^{2} + 1 = 0$

$λ = \pm i$

Eigenvalues: i, -i (Complex)

Example 5: Let $A = [a b b a] .$ Show that the eigenvalues of A are a + b and a - b.

Solution:

$A = [a b b a] \Rightarrow A - λ I = [a - λ b b a - λ]$

$det (A - λ I) = (a - λ)^{2} - b^{2} = 0$

$\Rightarrow (a - λ)^{2} = b^{2} \Rightarrow a - λ = \pm b \Rightarrow λ = a \pm b$

Eigenvalues: $a + b, a - b$

Example 6: Let $A = [6223] .$ Find the eigenvalues and verify:

Sum of eigenvalues = Trace(A)
Product of eigenvalues = det(A)

Solution:

Characteristic equation:

$det (A - λ I) = 6 - λ 2 2 3 - λ = 0$

$\Rightarrow (6 - λ) (3 - λ) - 4 = 0$

$\Rightarrow λ^{2} - 9 λ + 14 = 0$

$\Rightarrow λ = \frac{9 \pm 81 - 56}{2} = \frac{9 \pm 5}{2}$

$\Rightarrow λ_{1} = 7, λ_{2} = 2$

Verification:

Sum = 7 + 2 = 9 = Trace(A)
Product = 7⋅2 = 14 = det(A)

Verified

6.0Applications of Eigenvalues

Physics: Quantum mechanics, vibration analysis
Mathematics: Stability of systems, differential equations
Engineering: Stress-strain analysis
Data Science: Principal Component Analysis (PCA)

7.0Practice Question on Eigenvalues

Find the eigen values of the matrix: $A = [1221] .$ (Hint: Use the characteristic equation: $d e t (A - λ I) = 0$ )

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Eigenvalues of a Matrix

In linear algebra, eigenvalues are special numbers associated with a square matrix. They reveal important characteristics like stability, direction, and scaling of vector transformations. Widely used in mathematics, physics, and data science, eigenvalues simplify complex matrix operations. Understanding eigen values is key for solving systems of equations, quantum mechanics, and even in machine learning techniques like PCA (Principal Component Analysis).

1.0Definition of Eigenvalues

An eigenvalue of a square matrix A is a scalar λ such that:

$A x = λ x$

Here, A is an n × n matrix, x is a non-zero vector (called the eigenvector), and λ is the eigenvalue. This equation means that multiplying a matrix A by a vector x results in a scaled version of x, not a rotated or otherwise transformed one.

2.0Eigenvalue Formula

To find the eigenvalues of a matrix, use the characteristic equation:

$d e t (A - λ l) = 0$

Where:

A is a square matrix,
I is the identity matrix of the same size,
λ represents eigenvalues,
det stands for determinant.

This results in a polynomial equation in λ. The roots of this polynomial are the eigenvalues of the matrix.

3.0Eigenvalue of a 2×2 Matrix

For a 2×2 matrix: $A = [a c b d]$

The characteristic equation becomes:

$det ([a - λb c d - λ]) = 0$

$\Rightarrow (a - λ) (d - λ) - b c = 0$

$\Rightarrow λ^{2} - (a + d) λ + (a d - b c) = 0$

The roots of this quadratic equation give the eigenvalues of the 2×2 matrix.

4.0Properties of Eigen Values

Here are some key properties of eigen values for square matrices:

Only Square Matrices Have Eigenvalues: Non-square matrices do not have eigenvalues.
Sum of Eigenvalues = Trace of Matrix: $\sum λ_{1} = T r (A)$
Product of Eigenvalues = Determinant of Matrix: $\prod λ_{i} = det (A)$
Eigenvalues of a Diagonal Matrix are the Diagonal Elements.
If A is symmetric, all eigenvalues are real.
Eigenvalues of A⁻¹ are reciprocal of A’s eigenvalues.
If A is triangular, eigenvalues are its diagonal entries.

5.0Example: Finding Eigenvalues of a 2×2 Matrix

Example 1: Let $A = [4123]$

Solution:

Step 1: Set up the characteristic equation:

$det (A - λ I) = 0$

$= det ([4 - λ 1 2 3 - λ])$

$= (4 - λ) (3 - λ) - (2) (1)$

$= λ^{2} - 7 λ + 10 = 0$

Step 2: Solve the quadratic equation:

$λ = \frac{7 \pm 49 - 40}{2} = \frac{7 \pm 3}{2}$

$\Rightarrow λ_{1} = 5, λ_{2} = 2$

Hence, the eigenvalues are 5 and 2.

Example 2: Let $A = 2 - 1 0 - 1 2 - 1 0 - 1 2 .$ Find the eigenvalues of matrix A.

Solution:

This is a symmetric tridiagonal matrix. Use the characteristic equation:

$det (A - λ I) = 0$

$\Rightarrow 2 - λ - 1 0 - 1 2 - λ - 1 0 - 1 2 - λ = 0$

Let’s evaluate the determinant using cofactor expansion:

Let $D (λ) = det (A - λ I)$

Use shortcut tridiagonal determinant formula or compute directly:

$D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (- 1)^{2} (2 - λ) D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (2 - λ) D (λ) = (2 - λ) [(2 - λ)^{2} - 1] - (2 - λ)$

Factor out : $(2 - λ)$

$= (2 - λ) [(2 - λ)^{2} - 1 - 1] = (2 - λ) [(2 - λ)^{2} - 2]$

Let $x = 2 - λ$

$\Rightarrow D = x (x^{2} - 2) = x^{3} - 2 x = 0 \Rightarrow x (x^{2} - 2) = 0 \Rightarrow x = 0, x = \pm 2$

So,

$2 - λ = 0 \Rightarrow λ = 22 - λ = 2 \Rightarrow λ = 2 - 22 - λ = - 2 \Rightarrow λ = 2 + 2$

Eigenvalues:

$2, 2 + 2, 2 - 2$

Example 3: Find the eigenvalues of $A = 700 0 - 3 0 005 .$

Solution:

For a diagonal matrix, the eigenvalues are the diagonal elements.

So, the eigenvalues are:

7, −3, 5

Example 4: Let $A = [01 - 1 0] .$ Find the eigenvalues of A.

Solution:

$det (A - λ I) = - λ 1 - 1 - λ = λ^{2} + 1 = 0$

$λ = \pm i$

Eigenvalues: i, -i (Complex)

Example 5: Let $A = [a b b a] .$ Show that the eigenvalues of A are a + b and a - b.

Solution:

$A = [a b b a] \Rightarrow A - λ I = [a - λ b b a - λ]$

$det (A - λ I) = (a - λ)^{2} - b^{2} = 0$

$\Rightarrow (a - λ)^{2} = b^{2} \Rightarrow a - λ = \pm b \Rightarrow λ = a \pm b$

Eigenvalues: $a + b, a - b$

Example 6: Let $A = [6223] .$ Find the eigenvalues and verify:

Sum of eigenvalues = Trace(A)
Product of eigenvalues = det(A)

Solution:

Characteristic equation:

$det (A - λ I) = 6 - λ 2 2 3 - λ = 0$

$\Rightarrow (6 - λ) (3 - λ) - 4 = 0$

$\Rightarrow λ^{2} - 9 λ + 14 = 0$

$\Rightarrow λ = \frac{9 \pm 81 - 56}{2} = \frac{9 \pm 5}{2}$

$\Rightarrow λ_{1} = 7, λ_{2} = 2$

Verification:

Sum = 7 + 2 = 9 = Trace(A)
Product = 7⋅2 = 14 = det(A)

Verified

6.0Applications of Eigenvalues

Physics: Quantum mechanics, vibration analysis
Mathematics: Stability of systems, differential equations
Engineering: Stress-strain analysis
Data Science: Principal Component Analysis (PCA)

7.0Practice Question on Eigenvalues

Find the eigen values of the matrix: $A = [1221] .$ (Hint: Use the characteristic equation: $d e t (A - λ I) = 0$ )

Frequently Asked Questions

What is the difference between an eigenvalue and an eigenvector?

Can eigenvalues be complex numbers?

Are eigenvalues always real for symmetric matrices?

Frequently Asked Questions

What is the difference between an eigenvalue and an eigenvector?

Can eigenvalues be complex numbers?

Are eigenvalues always real for symmetric matrices?

Join ALLEN!

Eigenvalues of a Matrix

1.0Definition of Eigenvalues

2.0Eigenvalue Formula

3.0Eigenvalue of a 2×2 Matrix

4.0Properties of Eigen Values

5.0Example: Finding Eigenvalues of a 2×2 Matrix

6.0Applications of Eigenvalues

7.0Practice Question on Eigenvalues

Table of Contents

Join ALLEN!

Eigenvalues of a Matrix

1.0Definition of Eigenvalues

2.0Eigenvalue Formula

3.0Eigenvalue of a 2×2 Matrix

4.0Properties of Eigen Values

5.0Example: Finding Eigenvalues of a 2×2 Matrix

6.0Applications of Eigenvalues

7.0Practice Question on Eigenvalues

Table of Contents