Definition of applying a polynomial to a linear transformation
Created almost 9 years ago, updated 5 months ago
ID24048
Definition: If (p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k) is a polynomial and (T: V o V) is a linear transformation, then (p(T)) is the linear transformation: [p(T) = a_0 I + a_1 T + a_2 T^2 + \cdots + a_k T^k]
where (T^k) denotes the (k)-fold composition (T \circ T \circ \cdots \circ T) and (I) is the identity transformation.
Example: If (p(x) = x^2 - 3x + 2) and (T) is a linear operator, then: [p(T) = T^2 - 3T + 2I]
and for any vector (\mathbf{v}): (p(T)(\mathbf{v}) = T(T(\mathbf{v})) - 3T(\mathbf{v}) + 2\mathbf{v}).
**Definition:** If \(p(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_k x^k\) is a polynomial and \(T: V o V\) is a linear transformation, then \(p(T)\) is the linear transformation:
\[p(T) = a_0 I + a_1 T + a_2 T^2 + \cdots + a_k T^k\]
where \(T^k\) denotes the \(k\)-fold composition \(T \circ T \circ \cdots \circ T\) and \(I\) is the identity transformation.
**Example:** If \(p(x) = x^2 - 3x + 2\) and \(T\) is a linear operator, then:
\[p(T) = T^2 - 3T + 2I\]
and for any vector \(\mathbf{v}\): \(p(T)(\mathbf{v}) = T(T(\mathbf{v})) - 3T(\mathbf{v}) + 2\mathbf{v}\).