Eigenvector

Definition

An eigenvector is a non-zero vector that, when a linear operator acts on it, results in a scalar multiple of itself, which means it retains its direction while possibly changing its magnitude. This key property makes eigenvectors essential in various applications like solving systems of linear equations and transforming vector spaces. They are intrinsically linked to eigenvalues, which determine the scaling factor associated with each eigenvector.

5 Must Know Facts For Your Next Test

1. Eigenvectors can exist in any vector space, but they often reveal important structural properties of the space when analyzed in conjunction with their corresponding eigenvalues.
2. The set of all eigenvectors corresponding to a particular eigenvalue spans an eigenspace, which can have one or more dimensions depending on the multiplicity of the eigenvalue.
3. For any linear operator represented by a matrix, the characteristic polynomial can be used to find its eigenvalues, from which eigenvectors can be derived.
4. Eigenvectors are crucial in quantum mechanics as they represent measurable quantities, where the outcomes of measurements correspond to the eigenvalues.
5. Not every vector is an eigenvector; only those that satisfy the equation $$A extbf{v} = ext{λ} extbf{v}$$ for some scalar $$ ext{λ}$$ (the eigenvalue) qualify.

Review Questions

• How do eigenvectors and eigenvalues relate to each other in the context of linear transformations?
  • Eigenvectors and eigenvalues are fundamentally linked in that every eigenvector has a corresponding eigenvalue that indicates how the vector scales under a linear transformation. When a linear operator acts on an eigenvector, it produces a new vector that is simply a scaled version of the original. The relationship is captured by the equation $$A extbf{v} = ext{λ} extbf{v}$$, where $$A$$ is the operator, $$ extbf{v}$$ is the eigenvector, and $$ ext{λ}$$ is the eigenvalue. This relationship showcases how certain directions remain invariant under transformation while being scaled by their respective values.
• Discuss the significance of finding eigenvectors and their corresponding eigenvalues when diagonalizing a matrix.
  • Finding eigenvectors and their corresponding eigenvalues is crucial for diagonalizing a matrix because it simplifies the representation of linear transformations. By expressing a matrix in diagonal form using its eigenvalues and eigenvectors, complex operations such as raising matrices to powers or computing exponentials become significantly easier. Diagonalization reveals important structural insights about the matrix, allowing for more straightforward computations and interpretations in applications like stability analysis and quantum mechanics.
• Evaluate how the concept of eigenvectors applies to quantum mechanics and the measurement problem.
  • In quantum mechanics, eigenvectors play a central role in understanding observable quantities and their measurements. Each observable corresponds to a linear operator whose eigenvectors represent possible states that yield definite measurement outcomes. The associated eigenvalues denote these outcomes. When a measurement is made, the system collapses into one of these eigenstates with probabilities determined by their respective coefficients in the wave function's expansion. This interpretation illustrates how eigenvectors are not just mathematical tools but foundational elements in explaining physical phenomena in quantum theory.

Related terms

Eigenvalue:

An eigenvalue is a scalar associated with an eigenvector, indicating how much the eigenvector is stretched or compressed during the transformation by a linear operator.

Linear Operator: A linear operator is a mapping between vector spaces that preserves the operations of vector addition and scalar multiplication.

Diagonalization: Diagonalization is the process of converting a matrix into a diagonal form, simplifying the representation of linear transformations and making it easier to compute powers and exponentials of matrices.