Eigenvalue

An Eigenvalue is a real number, $\lambda$ that when multiplied with an eigenvector yields the original vector.

• AKA: Characteristic Root, Latent Value, Proper Value, Spectral Value.
• Context:
• It can be produced by an Eigenvalue Finding Task.
• It be represented as:
Let $V$ be a vector space over the field $F$ and let $T$ be a linear operator on $V$. An eigenvalue of $T$ is a scalar $\lambda$ in $F$ such that there is a non-zero vector $X$ in $V$ with $TX=\lambda X$.Here $X$ is called the eigen vector of T for respective eigen value $\lambda$.
• It can used in a Dimension-Compression Task (along with its eigenvectors).
• When a beam is struck, its natural frequency ( eigenvalues) can be measured. So eigenvalues can be used to test for cracks and deformities in the structural components used for construction.
• If $\lambda$ is an eigenvalue of a matrix $A$ and $\alpha$ be any scalar then
• The matrix $\alpha A$ has eigenvalue $\alpha \lambda$.
• The matrix $A^m$ has eigenvalue $\lambda^m$.
• The matrix $A-kI$ has eigenvalue $\lambda-k$.
• The matrix $A^{-1}$ has eigenvalue $\frac{1}{\lambda}$.
• The matrix $A$ and $A^T$ have same eigenvalues.
• Example(s):
• $\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 5 & 4 \\ 1 & 2 \end{bmatrix} \Bigr) = \{6,1\}$.
• $\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix} \Bigr) = \{ 1+2i , 1-2i \}$.
• $\operatorname{Eigenvalues} \Bigl(\begin{bmatrix} -2 & 2 & 3 \\ 2 & 1 & -6 \\ -1 & -2 & 0 \end{bmatrix}\Bigr) = \{ 5, -3 , -3 \}$ .
• See: Matrix Equation, Square Matrix.

References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Eigenvalues_and_eigenvectors Retrieved:2015-2-16.
• An eigenvector or characteristic vector of a linear transformation defines a direction that is invariant under the transformation. Let the transformation be defined by the square matrix A, then an invariant direction of A is the non-zero vector v, that has the property that the product Av is a scalar multiple of v. This is written as the equation, : $A\mathbf{v} = \lambda \mathbf{v},$ where λ is known as the eigenvalue associated with the eigenvector v.

(Because this equation uses post-multiplication of the matrix A by the vector v it describes an right eigenvector.) The number λ is called the eigenvalue or characteristic value of A corresponding to v.[1]

1. Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.

2013

• http://en.wikipedia.org/wiki/Eigenvalue
• An eigenvector of a square matrix $A$ is a non-zero vector $v$ that, when multiplied by $A$, yields the original vector multiplied by a single number $\lambda$; that is: $A v = \lambda v$ The number $\lambda$ is called the eigenvalue of $A$ corresponding to $v$. … Thus, for example, the exponential function $f(x) = a^x$ is an eigenfunction of the derivative operator " ${}'$ ", with eigenvalue $\lambda = \ln a$, since its derivative is $f'(x) = (\ln a)a^x = \lambda f(x)$.

The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix. An eigenspace of a matrix $A$ is the set of all eigenvectors with the same eigenvalue, together with the zero vector. An eigenbasis for $A$ is any basis for the set of all vectors that consists of linearly independent eigenvectors of $A$. Not every real matrix has real eigenvalues, but every complex matrix has at least one complex eigenvalue.