multiplicity of eigenvalues
(1 hours to learn)
The characteristic polynomial of a matrix may have repeated roots. If this is the case, the geometric multiplicity of a given eigenvalue (the dimension of the corresponding eigenspace) may be less than the algebraic multiplicity. This property determines whether a matrix is diagonalizable, and it is relevant to the solutions of differential equations.
This concept has the prerequisites:
- eigenvalues and eigenvectors
- roots of polynomials (The Fundamental Theorem of Algebra implies the sum of the algebraic multiplicities is the dimension of the space.)
Core resources (read/watch one of the following)
→ A First Course in Linear Algebra (2012)
A linear algebra textbook with proofs.
Location: Section "Properties of eigenvalues and eigenvectors," subsection "Multiplicities of eigenvalues"
→ Multivariable Mathematics
A textbook on linear algebra and multivariable calculus with proofs.
Location: Section 9.2, "Eigenvalues, eigenvectors, and applications," subsection 9.2.2, "Diagonalizability," pages 429-433
-No Additional Notes-
- create concept: shift + click on graph
- change concept title: shift + click on existing concept
- link together concepts: shift + click drag from one concept to another
- remove concept from graph: click on concept then press delete/backspace
- add associated content to concept: click the small circle that appears on the node when hovering over it
- other actions: use the icons in the upper right corner to optimize the graph placement, preview the graph, or download a json representation