How to find the basis of an eigenspace?
Understand the Problem
The question is asking for a method or procedure to determine the basis of an eigenspace in linear algebra. This typically involves finding eigenvalues and corresponding eigenvectors of a matrix, then expressing the eigenspace as a span of these eigenvectors.
Answer
The basis of an eigenspace is given by the span of its corresponding eigenvectors.
Answer for screen readers
The basis of an eigenspace can be expressed as the span of its corresponding eigenvectors found from the eigenvalues of the matrix.
Steps to Solve
- Find Eigenvalues Start by calculating the eigenvalues of the matrix $A$. To do this, you need to solve the characteristic polynomial given by the equation:
$$ \det(A - \lambda I) = 0 $$
where $\lambda$ represents the eigenvalue and $I$ is the identity matrix. This will result in a polynomial equation in $\lambda$.
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Solve Characteristic Polynomial Once you have the characteristic polynomial, find its roots. These roots are the eigenvalues of the matrix $A$.
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Find Eigenvectors For each eigenvalue $\lambda_i$, substitute it back into the equation:
$$ (A - \lambda_i I)v = 0 $$
where $v$ is the eigenvector corresponding to the eigenvalue $\lambda_i$. This will typically lead to a system of linear equations.
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Solve the System of Equations Use methods like Gaussian elimination or row reduction to solve the system of equations obtained in the previous step, which will give you the eigenvectors for each eigenvalue.
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Form the Eigenspace The eigenspace corresponding to each eigenvalue is the span of its eigenvectors. For eigenvalue $\lambda_i$ with eigenvectors $v_1, v_2, \ldots, v_k$, the eigenspace is given by:
$$ \text{Eigenspace}(\lambda_i) = \text{span}{v_1, v_2, \ldots, v_k} $$
- Repeat for All Eigenvalues Repeat steps 3 to 5 for all eigenvalues, gathering the eigenvectors for each eigenspace.
The basis of an eigenspace can be expressed as the span of its corresponding eigenvectors found from the eigenvalues of the matrix.
More Information
Once you find the eigenvalues and their corresponding eigenvectors, the set of all eigenvectors associated with an eigenvalue forms the eigenspace of that eigenvalue. The dimension of the eigenspace is known as the geometric multiplicity, which can provide insight into the matrix's characteristics, such as whether it is diagonalizable.
Tips
- Not checking the algebra when finding the characteristic polynomial.
- Forgetting to substitute correctly for each eigenvalue while finding eigenvectors.
- Confusing the eigenspace with just the eigenvector; remember it includes all linear combinations of the eigenvectors.
