Definition 5.1.2. Eigenvectors and Eigenvalues.
A vector \(\vec{x}\) is an \textit{eigenvector} of a matrix \(M\) if and only if \(A\vec{x}=\lambda\vec{x}\) for some number \(\lambda\) called the corresponding eigenvalue.
Section 5.1 Eigen Vector
Goals.
We will
- consider vectors with special properties with respect to a matrix,
- develop a method for finding these vectors, and
- discover a property of the set of these vectors.
We have considered matrices as transformations: they map vectors to new vectors. We are looking for vectors whose direction is not changed.
Subsection 5.1.1 Motivation
Guido is sailing with intended velocity \(\vec{x}\text{.}\) The effect of the wind on him is given by \(A\vec{x}\text{.}\) Find directions (\(\vec{x}\)) Guido can sail so that the wind is either directly at his back (helping him), or directly in his face (suggesting he turn around).
\begin{equation*}
A = \left[ \begin{array}{rr} 4 & -2 \\ -3 & 9 \end{array} \right].
\end{equation*}
We want to solve the following equation.
\begin{equation*}
A\vec{x}=\lambda \vec{x}
\end{equation*}
Because the vector is the same on both sides and \(\lambda\) is just a scalar, this equation says that the matrix does not change the direction of the vector. Note in this illustration \(\lambda\) indicates the magnitude of help; it is positive for a tailwind or negative for a headwind.
Solutions for \(A\) are \([-1,3]^T\) and \([2,1]^T\text{.}\) Note
\begin{equation*}
\left[ \begin{array}{rr} 4 & -2 \\ -3 & 9 \end{array} \right] \left[ \begin{array}{r} -1 \\ 3 \end{array} \right] = \left[ \begin{array}{r} -10 \\ 30 \end{array} \right] = 10\left[ \begin{array}{r} -1 \\ 3 \end{array} \right].
\end{equation*}
Also
\begin{equation*}
\left[ \begin{array}{rr} 4 & -2 \\ -3 & 9 \end{array} \right] \left[ \begin{array}{r} 2 \\ 1 \end{array} \right] = \left[ \begin{array}{r} 6 \\ 3 \end{array} \right] = 3\left[ \begin{array}{r} 2 \\ 1 \end{array} \right]
\end{equation*}
Note because scale does not matter if \(\vec{x}\) is a solution then so is \(c\vec{x}\text{.}\) If \(T\) is the linear transformation with matrix \(A\) these are vectors that are mapped onto the same line. View the results in the following images. For example \([6,3]^T\) also works.
If we apply this transformation to an image, points on these two vectors will be stretched but not rotated while the rest may also rotate.
Original image
Transformed image
Subsection 5.1.2 Method
First note the following, algebraic re-arrangement of the eigenvector definition.
\begin{align*}
A\vec{x} = & \lambda\vec{x}.\\
A\vec{x}-\lambda\vec{x} = & \vec{0}.\\
(A-\lambda I)\vec{x} = & \vec{0}.
\end{align*}
Activity 5.1.1.
Remember the eigenvectors of \(A\) are the solutions to \((A-\lambda I)\vec{x}=\vec{0}\text{.}\)
(a)
The set of all such \(\vec{x}\) is what with respect to the matrix \(A-\lambda I\text{?}\)
(b)
Given we want the non-trivial solutions to \((A-\lambda I)\vec{x}=\vec{0}\) what are we supposing is true of \(A-\lambda I\text{?}\)
(c)
Using the previous conclusion what do we know about \(\det(A-\lambda I)\text{?}\)
Checkpoint 5.1.3.
Find the eigenvalues and eigenvectors for \(B=\left[\begin{array}{rr} 5 & 0 \\ 2 & 1 \end{array} \right]\text{.}\)
Definition 5.1.4. Characteristic Equation.
The polynomial in \(\lambda\text{,}\) \(\det(A-\lambda I)\text{,}\) is called the characteristic equation of \(A\text{.}\)
Subsection 5.1.3 Independence of Eigenvectors
Next we consider the set of eigenvectors and ask if it will be independent.
Activity 5.1.2.
Consider eigenvectors \(\vec{v}_1, \vec{v}_2, \ldots, \vec{v}_n\) that correspond to distinct eigenvalues \(\lambda_1, \lambda_2, \ldots, \lambda_n\text{.}\)
Suppose \(\vec{v}_{p+1}\) is dependent on the independent eigenvectors \(\vec{v}_1, \ldots, \vec{v}_p\text{.}\)
(a)
Write \(c_1\vec{v}_1+c_2\vec{v}_2+\ldots+c_{n-1}\vec{v}_p=\vec{v}_{p+1}\text{.}\)
(b)
Multiply this equation by \(A\) and simplify remembering the definition of eigenvectors.
(c)
Again write \(c_1\vec{v}_1+c_2\vec{v}_2+\ldots+c_{n-1}\vec{v}_p=\vec{v}_{p+1}\text{.}\)
(d)
Multiply this copy by \(\lambda_{p+1}\text{.}\)
(e)
Subtract the two equations you generated.
(f)
Remembering that \(\vec{v}_1, \ldots, \vec{v}_p\) are independent, what does the resulting equation imply?
Theorem 5.1.5.
If the eigenvectors correspond to distinct eigenvalues, the eigenvectors are independent.
