Finding Eigenspace Bases For A Matrix A Step-by-Step Guide
In linear algebra, understanding eigenspaces and eigenvalues is crucial for analyzing the behavior of linear transformations. Eigenspaces provide a framework for decomposing a vector space into subspaces that are invariant under a given linear transformation. In this comprehensive guide, we will delve into the process of finding bases for eigenspaces, focusing on the matrix:
[ 1 0 0 ]
[ -8 4 -6 ]
[ 8 1 9 ]
This article will provide a step-by-step approach, ensuring a clear understanding of the underlying concepts and calculations. Let's embark on this journey to master the art of finding eigenspace bases.
Understanding Eigenvalues and Eigenvectors
Before we dive into the computational aspects, it's essential to grasp the fundamental concepts of eigenvalues and eigenvectors. An eigenvector of a matrix A is a non-zero vector v that, when multiplied by A, results in a scalar multiple of itself. This scalar is known as the eigenvalue, denoted by λ. Mathematically, this relationship is expressed as:
Av = λv*
In simpler terms, when a matrix A acts on an eigenvector v, it only scales the vector without changing its direction. The eigenvalue λ represents the factor by which the eigenvector is scaled. Eigenvalues and eigenvectors provide valuable insights into the behavior of a linear transformation represented by the matrix A. They reveal the directions in which the transformation acts purely by scaling, without any rotation or shearing.
The set of all eigenvectors corresponding to a particular eigenvalue, along with the zero vector, forms a subspace called the eigenspace. The eigenspace represents the set of all vectors that are scaled by the same factor (the eigenvalue) when the linear transformation is applied. Finding a basis for the eigenspace allows us to describe this subspace in a concise and meaningful way. A basis is a set of linearly independent vectors that span the eigenspace, meaning that any vector in the eigenspace can be expressed as a linear combination of the basis vectors.
Step 1: Finding the Eigenvalues
The first step in finding the bases for the eigenspaces is to determine the eigenvalues of the given matrix. For the matrix:
[ 1 0 0 ]
[ -8 4 -6 ]
[ 8 1 9 ]
We start by solving the characteristic equation, which is given by:
det(A - λI) = 0
where A is the given matrix, λ is the eigenvalue, and I is the identity matrix. Substituting the given matrix into the equation, we get:
det([ 1-λ 0 0 ])
[ -8 4-λ -6 ]) = 0
[ 8 1 9-λ ])
To compute the determinant of this 3x3 matrix, we can use the cofactor expansion method. Expanding along the first row, we have:
(1 - λ) * det([ 4-λ -6 ]) [ 1 9-λ ])
Calculating the determinant of the 2x2 matrix:
(4 - λ)(9 - λ) - (-6)(1) = (4 - λ)(9 - λ) + 6
Expanding this expression, we get:
36 - 4λ - 9λ + λ² + 6 = λ² - 13λ + 42
Now, substituting this back into the characteristic equation:
(1 - λ)(λ² - 13λ + 42) = 0
This is our characteristic polynomial. We can see that one eigenvalue is immediately λ = 1. To find the remaining eigenvalues, we need to solve the quadratic equation:
λ² - 13λ + 42 = 0
We can factor this quadratic equation as:
(λ - 6)(λ - 7) = 0
Thus, the eigenvalues are λ = 6 and λ = 7. Therefore, the eigenvalues for the given matrix are λ₁ = 1, λ₂ = 6, and λ₃ = 7.
In summary, to find eigenvalues, we formulate the characteristic equation by finding the determinant of (A - λI) and setting it to zero. Solving this equation gives us the eigenvalues of the matrix. The eigenvalues play a pivotal role in understanding the matrix's behavior and are essential for finding the eigenspaces.
Step 2: Finding the Eigenspace for λ = 1
Now that we have the eigenvalues, we can proceed to find the eigenspaces corresponding to each eigenvalue. Let's start with the eigenvalue λ₁ = 1. The eigenspace associated with λ₁ = 1 is the set of all vectors v that satisfy the equation:
(A - λ₁I)v = 0
Substituting λ₁ = 1 into this equation, we get:
(A - I)v = 0
where I is the 3x3 identity matrix. The matrix (A - I) is:
[ 1-1 0 0 ] [ 0 0 0 ]
[ -8 4-1 -6 ] = [ -8 3 -6 ]
[ 8 1 9-1 ] [ 8 1 8 ]
So, we need to solve the following homogeneous system of linear equations:
[ 0 0 0 ] [ x ] = [ 0 ]
[ -8 3 -6 ] [ y ] = [ 0 ]
[ 8 1 8 ] [ z ] = [ 0 ]
This corresponds to the following system of equations:
-8x + 3y - 6z = 0
8x + y + 8z = 0
To solve this system, we can use Gaussian elimination or row reduction. First, let's add the second equation to the first equation:
4y + 2z = 0
This simplifies to:
2y + z = 0
So, z = -2y. Now, substitute this into the second equation:
8x + y + 8(-2y) = 0
8x + y - 16y = 0
8x - 15y = 0
8x = 15y
x = (15/8)y
Now, let's express the solution in terms of a free variable. Let y = t. Then, x = (15/8)t and z = -2t. The solution vector v can be written as:
[ x ] [ (15/8)t ] [ 15/8 ]
[ y ] = [ t ] = t [ 1 ]
[ z ] [ -2t ] [ -2 ]
To eliminate the fraction, we can multiply the vector by 8, which gives us the eigenvector:
[ 15 ]
[ 8 ]
[ -16]
Thus, a basis for the eigenspace corresponding to λ₁ = 1 is the single vector:
{[ 15, 8, -16 ]}
The eigenspace associated with λ = 1 is a one-dimensional subspace spanned by this eigenvector. This means that any vector in this eigenspace can be obtained by scaling this basis vector.
In essence, finding the eigenspace involves solving a homogeneous system of linear equations derived from (A - λI)v = 0. The solutions to this system form the eigenspace, and a basis for this space provides a concise representation of all vectors that are scaled by the eigenvalue when multiplied by the matrix.
Step 3: Finding the Eigenspace for λ = 6
Next, we find the eigenspace corresponding to the eigenvalue λ₂ = 6. We need to solve the equation:
(A - λ₂I)v = 0
Substituting λ₂ = 6, we get:
(A - 6I)v = 0
The matrix (A - 6I) is:
[ 1-6 0 0 ] [ -5 0 0 ]
[ -8 4-6 -6 ] = [ -8 -2 -6 ]
[ 8 1 9-6 ] [ 8 1 3 ]
So, we need to solve the homogeneous system:
[ -5 0 0 ] [ x ] = [ 0 ]
[ -8 -2 -6 ] [ y ] = [ 0 ]
[ 8 1 3 ] [ z ] = [ 0 ]
This corresponds to the following system of equations:
-5x = 0
-8x - 2y - 6z = 0
8x + y + 3z = 0
From the first equation, we have x = 0. Substituting x = 0 into the second and third equations, we get:
-2y - 6z = 0
y + 3z = 0
8(0) + y + 3z = 0
Both equations are essentially the same, which simplifies to:
y + 3z = 0
y = -3z
Now, let's express the solution in terms of a free variable. Let z = t. Then, y = -3t and x = 0. The solution vector v can be written as:
[ x ] [ 0 ]
[ y ] = [ -3t ] = t [ -3 ]
[ z ] [ t ] [ 1 ]
Thus, a basis for the eigenspace corresponding to λ₂ = 6 is the single vector:
{[ 0, -3, 1 ]}
The eigenspace associated with λ = 6 is a one-dimensional subspace spanned by this eigenvector.
In summary, the process for λ = 6 mirrors that of λ = 1. We form the matrix (A - λI), solve the resulting homogeneous system, and express the solution in terms of a free variable to find the basis vector for the eigenspace.
Step 4: Finding the Eigenspace for λ = 7
Finally, we find the eigenspace corresponding to the eigenvalue λ₃ = 7. We need to solve the equation:
(A - λ₃I)v = 0
Substituting λ₃ = 7, we get:
(A - 7I)v = 0
The matrix (A - 7I) is:
[ 1-7 0 0 ] [ -6 0 0 ]
[ -8 4-7 -6 ] = [ -8 -3 -6 ]
[ 8 1 9-7 ] [ 8 1 2 ]
So, we need to solve the homogeneous system:
[ -6 0 0 ] [ x ] = [ 0 ]
[ -8 -3 -6 ] [ y ] = [ 0 ]
[ 8 1 2 ] [ z ] = [ 0 ]
This corresponds to the following system of equations:
-6x = 0
-8x - 3y - 6z = 0
8x + y + 2z = 0
From the first equation, we have x = 0. Substituting x = 0 into the second and third equations, we get:
-3y - 6z = 0
y + 2z = 0
8(0) + y + 2z = 0
Both equations are essentially the same, which simplifies to:
y + 2z = 0
y = -2z
Now, let's express the solution in terms of a free variable. Let z = t. Then, y = -2t and x = 0. The solution vector v can be written as:
[ x ] [ 0 ]
[ y ] = [ -2t ] = t [ -2 ]
[ z ] [ t ] [ 1 ]
Thus, a basis for the eigenspace corresponding to λ₃ = 7 is the single vector:
{[ 0, -2, 1 ]}
The eigenspace associated with λ = 7 is a one-dimensional subspace spanned by this eigenvector.
In conclusion, we follow the same procedure for λ = 7 as we did for the previous eigenvalues. We solve the homogeneous system derived from (A - λI)v = 0 to find the basis vector for the corresponding eigenspace.
Summary of Eigenspace Bases
We have successfully found the bases for the eigenspaces corresponding to each eigenvalue of the given matrix:
[ 1 0 0 ]
[ -8 4 -6 ]
[ 8 1 9 ]
Here's a summary of our findings:
- For λ₁ = 1, the eigenspace basis is {[15, 8, -16]}
- For λ₂ = 6, the eigenspace basis is {[0, -3, 1]}
- For λ₃ = 7, the eigenspace basis is {[0, -2, 1]}
Each eigenspace is a one-dimensional subspace, meaning it is spanned by a single eigenvector. These eigenvectors represent the directions in which the matrix acts purely by scaling, without any rotation or shearing. The corresponding eigenvalues represent the scaling factors.
In summary, finding the eigenspaces and their bases involves calculating eigenvalues by solving the characteristic equation det(A - λI) = 0, and then for each eigenvalue, solving the homogeneous system (A - λI)v = 0 to find the eigenvectors that form the basis of the eigenspace.
Conclusion
In this guide, we have provided a comprehensive step-by-step approach to finding bases for the eigenspaces of a given matrix. We started by understanding the fundamental concepts of eigenvalues and eigenvectors, then moved on to the computational steps. We calculated the eigenvalues by solving the characteristic equation, and for each eigenvalue, we found the corresponding eigenspace by solving a homogeneous system of linear equations.
Understanding eigenspaces is crucial in various applications of linear algebra, including diagonalization, solving systems of differential equations, and analyzing the stability of systems. By mastering the techniques presented in this guide, you will be well-equipped to tackle problems involving eigenvalues and eigenvectors.
In conclusion, the process of finding eigenspace bases is a cornerstone of linear algebra. It provides a powerful tool for analyzing linear transformations and understanding the behavior of matrices. By diligently following the steps outlined in this guide, you can confidently determine the eigenspaces and their bases for any given matrix, unlocking deeper insights into the underlying linear transformations.
