Editing Square matrix (section)

===Eigenvalues and eigenvectors===
{{Main|Eigenvalue, eigenvector and eigenspace|l1=Eigenvalues and eigenvectors}}
A number {{mvar|λ}} and a non-zero vector <math>\mathbf{v}</math> satisfying
<math display="block">A \mathbf{v} = \lambda \mathbf{v}</math>
are called an ''eigenvalue'' and an ''eigenvector'' of {{nowrap|<math>A</math>,}} respectively.<ref>''Eigen'' means "own" in [[German language|German]] and in [[Dutch language|Dutch]].</ref><ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.1 }}</ref> The number {{mvar|λ}} is an eigenvalue of an {{math|''n''×''n''}}-matrix {{mvar|A}} if and only if {{math|''A'' − λ''I''<sub>''n''</sub>}} is not invertible, which is [[logical equivalence|equivalent]] to<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Definition III.4.9 }}</ref>
<math display="block">\det(A-\lambda I) = 0.</math>
The polynomial {{math|''p''<sub>''A''</sub>}} in an [[indeterminate (variable)|indeterminate]] {{math|''X''}} given by evaluation of the determinant {{math|det(''XI''<sub>''n''</sub> − ''A'')}} is called the [[characteristic polynomial]] of {{mvar|A}}. It is a [[monic polynomial]] of [[degree of a polynomial|degree]] ''n''. Therefore the polynomial equation {{math|1=''p''<sub>''A''</sub>(λ) = 0}} has at most ''n'' different solutions, i.e., eigenvalues of the matrix.<ref>{{Harvard citations |last1=Brown |year=1991 |nb=yes |loc=Corollary III.4.10 }}</ref> They may be complex even if the entries of {{mvar|A}} are real. According to the [[Cayley–Hamilton theorem]], {{math|1=''p''<sub>''A''</sub>(''A'') = 0}}, that is, the result of substituting the matrix itself into its own characteristic polynomial yields the [[zero matrix]].