Eigenvalues and Eigenvectors

Eigenvalues and eigenvectors are among the most important concepts in linear algebra. An eigenvector of a matrix is a non-zero vector whose direction is unchanged (or reversed) by the transformation — only its magnitude is scaled. This scaling factor is the eigenvalue.

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Definitions

For a square matrix $A$A, a non-zero vector $\mathbf{v}$v is an eigenvector of $A$A if:

$A\mathbf{v} = \lambda\mathbf{v}$Av=λv

for some scalar $\lambda$λ. The scalar $\lambda$λ is the corresponding eigenvalue.

Geometrically: $A$A transforms $\mathbf{v}$v into a scalar multiple of itself. The eigenvector lies on an invariant line through the origin.

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Finding Eigenvalues

Rearranging $A\mathbf{v} = \lambda\mathbf{v}$Av=λv:

$(A - \lambda I)\mathbf{v} = \mathbf{0}$(A−λI)v=0

For a non-zero solution $\mathbf{v}$v to exist, the matrix $A - \lambda I$A−λI must be singular:

$\det(A - \lambda I) = 0$det(A−λI)=0

This is called the characteristic equation. For an $n \times n$n×n matrix, it is a polynomial of degree $n$n in $\lambda$λ.

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2×2 Example

Worked Example 1: Find the eigenvalues and eigenvectors of $A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$A=(42​13​).

Step 1: Characteristic equation.