Eigenspace, eigenvalue, linear transformation, eigenvectors are four central concepts related to computing a basis of the eigenspace corresponding to a given eigenvalue. An eigenspace is a subspace of a vector space consisting of all eigenvectors corresponding to a specific eigenvalue. The eigenvalue is a scalar value associated with an eigenvector that remains unchanged when the linear transformation is applied to the eigenvector. The linear transformation is a function that maps one vector space to another, and the eigenvectors are nonzero vectors that are transformed by the linear transformation by a factor equal to the eigenvalue.
Finding a Basis for the Eigenspace of an Eigenvalue
Consider the matrix A and an eigenvalue $\lambda$ of A. The corresponding eigenspace $E_\lambda$ is the set of all vectors $\mathbf{x}$ that satisfy $A\mathbf{x} = \lambda \mathbf{x}$. To find a basis for $E_\lambda$, we can follow these steps:
1. Solve the System of Equations:
- Find a system of linear equations $(A – \lambda I)\mathbf{x} = \mathbf{0}$, where I is the identity matrix.
- Solve the system for the vector $\mathbf{x}$ (eigenvector).
2. Repeat for Other Eigenvectors:
- If there are multiple eigenvectors corresponding to $\lambda$, repeat steps 1 and 2 for each eigenvector.
- Each eigenvector will form a basis vector for $E_\lambda$.
3. Check for Linear Independence:
- The found eigenvectors must be linearly independent to form a basis.
- If they are not, compute linear combinations to find a set of linearly independent vectors.
4. Normalize Vectors (Optional):
- To simplify calculations, you may normalize the eigenvectors by dividing each by its length.
- This allows you to work with orthonormal basis vectors.
Example:
Let A = [[2, 1], [-1, 2]]. Find the basis for the eigenspace of $\lambda = 3$.
| Step 1: Solve (A – 3I)x = 0 |
|---|
| [[2, 1], [-1, 2]] – [[3, 0], [0, 3]] = [[-1, 1], [-1, -1]] |
| Equations: -x + y = 0, -x – y = 0 |
| Solution: x = y |
| Step 2: Eigenvector |
| Eigenvector: [1, 1] |
As there is only one linearly independent vector, it forms the basis for the eigenspace of $\lambda = 3$.
Question 1:
How to compute a basis for the eigenspace corresponding to an eigenvalue?
Answer:
To compute a basis for the eigenspace corresponding to an eigenvalue λ of a square matrix A, follow these steps:
- Solve the homogeneous linear system (A – λI)x = 0, where I denotes the identity matrix.
- The solutions to this system form a subspace called the eigenspace corresponding to λ.
- Choose a set of linearly independent vectors from the eigenspace to form a basis.
Question 2:
What is the importance of finding the eigenspace corresponding to an eigenvalue?
Answer:
The eigenspace corresponding to an eigenvalue λ represents the set of all linear combinations of eigenvectors associated with λ. This eigenspace is important for understanding the geometric properties of the matrix A:
- It provides information about the direction in which vectors are stretched or shrunk under the transformation represented by A.
- It helps in diagonalizing A, which simplifies matrix computations and provides insights into the matrix’s behavior.
Question 3:
How to determine whether a matrix has linearly independent eigenvectors for distinct eigenvalues?
Answer:
To determine whether a matrix has linearly independent eigenvectors for distinct eigenvalues, check if each eigenvalue has a corresponding eigenvector that is linearly independent of all eigenvectors corresponding to other eigenvalues. If this condition holds for all distinct eigenvalues, then the matrix has linearly independent eigenvectors.
- A matrix has linearly independent eigenvectors for distinct eigenvalues if it is diagonalizable.
- If the matrix is not diagonalizable, then it may or may not have linearly independent eigenvectors for distinct eigenvalues.
Thanks for sticking with me through this journey of linear algebra. I know it can be a bit daunting at first, but once you get the hang of it, it’s actually pretty cool. Seriously, who would have thought that there’s actually a systematic way to find all the special solutions to a system of equations? That’s like magic!
Anyway, I hope you found this article helpful. If you did, please consider sharing it with your friends or classmates. And if you have any questions, feel free to leave a comment below. I’ll do my best to answer them.
See you next time!