Eigenvectors and Eigenvalues

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In linear algebra, when a transformation of a space is carried out, some vectors (points in the space) are not rotated, but only extended or shrunk (moved farther or closer to the origin). The vectors are called the eigenvectors of the transformation, and the amount of extension or shrinkage carried out on that eigenvector is called the eigenvalue of the transformation corresponding to the given eigenvector.

Characteristic Property of Eigenvalues and Eigenvectors

An eigenvalue of a square n × n matrix with real entries A is a scalar \lambda \in \mathbb{R} such that

A\boldsymbol{x}=\lambda\boldsymbol{x}

for some non-zero vector \boldsymbol{x}\in\mathbb{R}^n known as a eigenvector. The eigenvalues are the zeroes of a matrix's characteristic polynomial, the degree of the corresponding root is called the algebraic multiplicity of the eigenvalue.

The same definition is valid for n × n matrices over any field F: Then \lambda \in \mathbf{F} and \boldsymbol{x}\in\mathbf{F}^n.

The eigenvectors represent directions that are preserved by linear transformations of a vector space.

All the eigenvectors of a particular eigenvalue span a vector space called the eigenspace.

If the characteristic polynomial splits into linear factors, then he product of all the eigenvalues of a matrix counted with their algebraic multiplicities equals the value of the matrix's determinant. Since a matrix is invertible if and only if the determinant is non-zero, it is invertible if and only if zero is not an eigenvalue.

The span of all the eigenvectors corresponding to a fixed eigenvalue λ is called the eigenspace Eλ of A. The dimension of this space is called the geometric multiplicity of the eigenvalue.

Eigenvectors and Eigenvalues in Physics

In Physics, eigenvectors and eigenvalues play an important role. For example, solving Schrödinger's equation (one of the fundamental equations of Quantum mechanics) is a problem of finding Eigenvectors and Eigenvalues for a very complex linear operator. The quantization (which means that discrete values are acceptable in physics, but not all of them: for example, an electron can have spin 1/2 or -1/2, but never 1/3 or 1/10) that was previously a postulated of Quantum mechanics can be derived from the fact that the eigenvalues of the operators come in discrete sets.

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