Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors of a Linear Transformation describe the change in the system. The eigenvalues are denoted by the symbol $\lambda$ , which relates to the magnitude of the transformation. Whereas, the eigenvector represents the way of transformation, such as a rotation or change in size.

A visual representation of a transformation can be seen below. Let's say we start with a unit square, dimensions 1-by-1. Now after applying a specific transformation the unit square deforms into the green parallelogram. The way of transforming an object by a rotation or magnification is described in the eigenvector. The size of this transformation is stored in the eigenvalue.

Visualization of the eigenvalues and eigenvectors

Mathematically a linear transformation can be described by a n-by-n matrix A. For the matrix the following formula can be used for solving for the eigenvalues and eigenvectors.

\begin{equation} Av = \lambda v \end{equation}

with the transformation matrix A, the eigenvalues $\lambda$ and eigenvectors $v$ .

Eigenvalues $\lambda$

If we want to solve for the eigenvalues $\lambda$ equation (1) should be stated as

\begin{equation} (A-\lambda I)v = 0 \end{equation}

where $I$ is the Identity-matrix and $0$ the zero vector both with dimensions n-by-n.

There only exists a nonzero solution for the eigenvectors $v$ if and only if the determinant of equation 2 is zero. Therefore, equation 2 can be rewritten into equation 3.

\begin{equation} \det(A-\lambda I) = 0 \end{equation}

Since A is a n-by-n matrix, the polynomial that results from equation 3 will also be of degree n. This polynomial is also called the characteristic polynomial. From the characteristic polynomial each eigenvalue can be computed by solving the polynomial. This is shown in the example at the end of this page.

Eigenvectors $v$

Now for each uniquely found eigenvalue $\lambda$ there exists an unique eigenvector $v$ . The eigenvectors can be computed by substituting each $\lambda_i$ into equation 2 and by knowing that the resulting system should be zero for every $v$ the eigenvectors can be found.

\begin{equation} (A-\lambda_i I) v_i = 0 \end{equation}

Example

A concise example is shown below for both computing the eigenvalues and eigenvectors. In this example the following transformation A matrix is considered

\mathbf{A} = \left[\begin{array} {cc} 2 & 1 \\ 1 & 2 \end{array}\right]

First the determinant of the transformation matrix a is computed this results into the characteristic polynomial as shown below.

\det(A-\lambda I) = \left[\begin{array} {cc} 2-\lambda & 1 \\ 1 & 2-\lambda \end{array}\right] = \lambda^2 - 4\lambda + 3

By solving the characteristic polynomial, n eigenvalues can be found namely

\lambda_1 = 1, \lambda_2 = 3

By substituting $\lambda_1$ and $\lambda_2$ into equation 4, the corresponding eigenvectors are found. As there can be seen a system is obtained form substituting the eigenvalues. Since there are two equations and also two unknowns the system can be solved by substitution.

Solving for $\lambda_1$

(A-I)v_{\lambda=1} = \left[\begin{array} {cc} 2-1 & 1 \\ 1 & 2-1 \end{array}\right]\left[\begin{array} {c} v_{1} \\ v_{2} \end{array}\right] = 0

v_1 + v_2 = 0

v_1 = -v_2

v_{\lambda=1} = \left[\begin{array} {c} v_{1} \\ -v_{1} \end{array}\right] = \left[\begin{array} {c} 1 \\ -1 \end{array}\right]

Solving for $\lambda_2$

(A-3I)v_{\lambda=3} = \left[\begin{array} {cc} 2-3 & 1 \\ 1 & 2-3 \end{array}\right]\left[\begin{array} {c} v_{1} \\ v_{2} \end{array}\right] = 0

-v_1 + v_2 = 0

v_1 - v_2 = 0

v_1 = v_2

v_{\lambda=3} = \left[\begin{array} {c} v_{1} \\ v_{2} \end{array}\right] = \left[\begin{array} {c} 1 \\ 1 \end{array}\right]

It is important to note that any other multiplicity of the eigenvector is also an eigenvector of the transformation matrix A. Such another multiplicity can be of the for

v_{\lambda=1} = \left[\begin{array} {c} 2 \\ -2 \end{array}\right]

Now by knowing this we can extend the usage of eigenvalues and eigenvectors. Interesting applications where this math can be applied to is for example in a dynamic analysis or for checking the stability of a system.

Eigenvalues and Eigenvectors

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Eigenvalues $\lambda$

Eigenvectors $v$

Example

More about Linear Algebra

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Determinant [2x2]

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