Eigenvectors and Eigenvalues

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.

Eigenvalues

An eigenvalue of a square matrix $A$ is a scalar $\lambda$ 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.

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 $\lambda$ is called the eigenspace $E_{\lambda }$ of $A$. The dimension of this space is called the geometric multiplicity of the eigenvalue.

Eigenvectors

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

An eigenvector of an n x n square matrix $A$ is a vector ${\boldsymbol {x}}\in \mathbb {C} ^{n}$ such that,

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

where $\lambda$ is complex number, known as a eigenvalue. The eigenvalue is the zero of a matrix's charateristic polynomial.

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