Eigenvalue and eigenspace
λ is an eigenvalue when ker(A−λI) is nonzero. That kernel is the eigenspace, including its zero vector.
Find a concept
REFERENCE SHEET
Eigenvalues & diagonalization
Find directions on which a linear transformation acts only by scaling.
Before this Vector spaces & linear maps, Polynomials & roots
Av = λv, v ≠ 0; A = PDP⁻¹
Definitions
Diagonalizable
A square matrix over F is diagonalizable over F if it has a basis of eigenvectors in Fⁿ.
Algebraic vs geometric multiplicity
Algebraic multiplicity is multiplicity in the characteristic polynomial; geometric multiplicity is the eigenspace dimension.
Key theorems
Distinct eigenvalues
Eigenvectors for distinct eigenvalues are linearly independent. In particular, n distinct eigenvalues imply diagonalizability of an n×n matrix.
Spectral theorem
Every real symmetric matrix has an orthonormal eigenbasis and real eigenvalues. Every complex Hermitian matrix has a unitary diagonalization with real eigenvalues.
Notable applications
Dynamics
Diagonalization simplifies powers Aᵏ and solutions of x′=Ax.
Principal components
Eigenvectors of a covariance matrix identify orthogonal directions of variance.
Work it out
Try before revealing01 / Compute
Diagonalize A=[[2,1],[1,2]] with an orthonormal basis.
Hint
Try vectors (1,1) and (1,−1).
Solution
The unit eigenvectors are (1,1)/√2 with λ=3 and (1,−1)/√2 with λ=1. Put these as columns of Q: A=Q diag(3,1) Qᵀ.
02 / Explain
Is [[1,1],[0,1]] diagonalizable?
Hint
Compute ker(A−I).
Solution
No. Its only eigenvalue is 1 and its eigenspace consists of (x,0), so it has dimension 1 rather than 2.
03 / Check an assumption
For A=[[1,1],[0,1]], compute Aⁿ for n≥1. Why does the eigenvalue alone miss part of the behavior?
Hint
Write A=I+N and check N².
Solution
N=[[0,1],[0,0]] has N²=0. The binomial expansion gives Aⁿ=I+nN=[[1,n],[0,1]]. Although the sole eigenvalue is 1, the off-diagonal term grows linearly because A has a nontrivial Jordan block.
Connect the ideas
Build on this
These sheets use this topic as a prerequisite.
Go deeper
Further reading and full-length treatments: