Eigenvalues are scalars describing how a matrix scales certain directions, while eigenvectors are the non-zero vectors in those directions. They satisfy , where is the matrix, is an eigenvector, and is its eigenvalue.
Eigenvalues are scalars describing how a matrix scales certain directions, while eigenvectors are the non-zero vectors in those directions. They satisfy , where is the matrix, is an eigenvector, and is its eigenvalue.
This is HL-only content in AHL 1.15: Eigenvalues and eigenvectors, applicable to all Mathematics AI HL paper styles. A GDC is available on every paper.
In IB terms, this idea identifies invariant directions under a linear transformation. It also supports diagonalization and later applications such as powers of matrices, population movement models, Markov chains, and phase portraits.
Geometrically, most vectors change both direction and length when transformed by a matrix. An eigenvector is exceptional: its direction remains on the same line, although a negative eigenvalue reverses its orientation. The eigenvalue gives the scale factor.
| Concept | Meaning |
|---|---|
| Eigenvector | A non-zero vector whose direction is unchanged by multiplication by |
| Eigenvalue | The scale factor applied to the corresponding eigenvector |
To find the eigenvalues, rewrite the defining equation:
A non-zero solution exists only when is singular, so . Once each eigenvalue is known, substitute it into and solve for a non-zero vector.
The determinant condition matters because a singular matrix has no inverse, allowing a non-zero solution to the homogeneous system. If the determinant were non-zero, the only solution would be the zero vector, which cannot be an eigenvector.
For example, let
The characteristic equation is
Therefore, or . For , the equations give , so an eigenvector is . For , they give , producing .
Check the first pair:
A common misconception is that each eigenvalue has exactly one eigenvector. Every non-zero scalar multiple is also an eigenvector; the zero vector is never allowed.
Exam technique: Show , pair each eigenvector with the correct eigenvalue, and verify using . For diagonalization, the columns of are eigenvectors and the matching eigenvalues form in . AHL 1.15 restricts this to matrices with distinct real eigenvalues.