# 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.