characteristic polynomial (original) (raw)

Characteristic Polynomial of a Matrix

| pA⁢(x):=det⁡(A-x⁢I)=|a11-xa12⋯a1⁢na21a22-x⋯a2⁢n⋮⋮⋱⋮an⁢1an⁢2⋯an⁢n-x| | | ------------------------------------------------------------------------- |

Characteristic Polynomial of a Linear Operator

Now, let T be a linear operator on a vector space V of dimension n<∞. Let α and β be any two ordered bases for V. Then we may form the matrices [T]α and [T]β. The two matrix representations of T are similar matrices, related by a change of bases matrix. Therefore, by the second remark above, we define the characteristic polynomial of T, denoted by pT⁢(x), in the indeterminate x, by

The characteristic equation of T is defined accordingly.