Eigenvector

Definition

An eigenvector is a non-zero vector that changes only by a scalar factor when a linear transformation is applied to it. This concept is crucial in understanding how linear operators behave, as eigenvectors correspond to specific directions in which these operators stretch or compress space. They are closely related to eigenvalues, which provide the scaling factor associated with each eigenvector, and they play a vital role in diagonalization, allowing matrices to be expressed in simpler forms that reveal their underlying structure.

5 Must Know Facts For Your Next Test

1. Eigenvectors must satisfy the equation $$A extbf{v} = extlambda extbf{v}$$, where A is a matrix, $$ extbf{v}$$ is the eigenvector, and $$ extlambda$$ is the corresponding eigenvalue.
2. An eigenvector can be scaled by any non-zero constant, and it will still be considered an eigenvector associated with the same eigenvalue.
3. The set of all eigenvectors corresponding to a given eigenvalue, along with the zero vector, forms a vector space known as the eigenspace.
4. Different linear transformations may have different sets of eigenvectors, and some matrices may have complex eigenvectors depending on their characteristics.
5. Finding the eigenvectors of a matrix often involves solving a system of equations derived from its characteristic polynomial.

Review Questions

• How do eigenvectors relate to linear transformations and their corresponding eigenvalues?
  • Eigenvectors are special vectors that remain in the same direction after a linear transformation is applied, merely being scaled by their corresponding eigenvalue. This relationship shows how linear transformations can stretch or compress space in specific directions represented by these vectors. In essence, while the transformation modifies the entire space, eigenvectors indicate directions that are invariant under this modification.
• Describe how you would go about finding the eigenvectors of a given matrix and explain why this process is significant.
  • To find the eigenvectors of a given matrix, you first need to calculate its eigenvalues by solving the characteristic polynomial equation $$ ext{det}(A - extlambda I) = 0$$. Once you have the eigenvalues, substitute them back into the equation $$A extbf{v} = extlambda extbf{v}$$ and solve for the vector $$ extbf{v}$$. This process is significant because it not only reveals key properties of the matrix but also helps in applications like stability analysis and simplifying systems of equations through diagonalization.
• Evaluate how the concept of eigenspaces expands our understanding of a matrix's behavior beyond just its individual eigenvectors.
  • Eigenspaces provide a broader perspective on how matrices behave by encompassing all possible eigenvectors associated with a specific eigenvalue along with the zero vector. This means that instead of just looking at individual vectors, we consider entire subspaces that are invariant under the transformation defined by the matrix. Understanding eigenspaces allows us to grasp complex dynamics such as stability and oscillation modes in systems modeled by these matrices, leading to deeper insights in applications ranging from engineering to quantum mechanics.

Related terms

Eigenvalue: An eigenvalue is a scalar associated with an eigenvector that indicates how much the eigenvector is stretched or compressed during the transformation.

Diagonalization: Diagonalization is the process of converting a matrix into a diagonal form, where its eigenvalues appear on the diagonal, simplifying many matrix operations.

Linear Transformation: A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.