Eigenvalue

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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 charateristic polynomial, the degree of the corresponding root is called the algebraic multiplicity of the eigenvalue.

The product of all the eigenvalues of a matrix 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.