Eigenspace

Dependencies:

1. Eigenvalues and Eigenvectors
2. Condition for being a subspace

Let $T: V \mapsto V$ be a linear operator. Let $\lambda$ be an eigenvalue of $T$.

Let $E_{\lambda} = \{x \in V: T(x) = \lambda x\}$. $E_{\lambda}$ is called the eigenspace for $(T, \lambda)$.

$E_{\lambda}$ is a subspace of $V$.

Proof

\[ T(0) = T(0+0) = T(0) + T(0) \implies T(0) = 0 = \lambda 0 \implies 0 \in E_{\lambda} \]

\[ x, y \in E_{\lambda} \implies T(x+y) = T(x) + T(y) = \lambda x + \lambda y = \lambda (x + y) \implies x + y \in E_{\lambda} \]

\[ x \in E_{\lambda} \implies T(cx) = cT(x) = c(\lambda x) = \lambda (cx) \implies cx \in E_{\lambda} \]

Since $E_{\lambda}$ is a non-empty subset of $V$ which is closed under addition and scalar multiplication, it is a subspace of $V$.

Dependency for: None

Info:

• Depth: 12
• Number of transitive dependencies: 57

Transitive dependencies:

1. /linear-algebra/vector-spaces/condition-for-subspace
2. /linear-algebra/matrices/gauss-jordan-algo
3. /sets-and-relations/composition-of-bijections-is-a-bijection
4. /sets-and-relations/equivalence-relation
5. Group
6. Ring
7. Polynomial
8. Integral Domain
9. Comparing coefficients of a polynomial with disjoint variables
10. Field
11. Vector Space
12. Linear independence
13. Zeros in vector space
14. Negation in vector space
15. Span
16. Linear transformation
17. Composition of linear transformations
18. Vector space isomorphism is an equivalence relation
19. Identity of a group is unique
20. Subgroup
21. Inverse of a group element is unique
22. Conditions for a subset to be a subgroup
23. Condition for a subset to be a subgroup
24. Condition for being a subspace
25. Semiring
26. Matrix
27. Stacking
28. System of linear equations
29. Product of stacked matrices
30. Matrix multiplication is associative
31. Reduced Row Echelon Form (RREF)
32. Matrices over a field form a vector space
33. Row space
34. Elementary row operation
35. Every elementary row operation has a unique inverse
36. Row equivalence of matrices
37. Row equivalent matrices have the same row space
38. RREF is unique
39. Identity matrix
40. Inverse of a matrix
41. Inverse of product
42. Elementary row operation is matrix pre-multiplication
43. Row equivalence matrix
44. Equations with row equivalent matrices have the same solution set
45. Basis of a vector space
46. Linearly independent set is not bigger than a span
47. Homogeneous linear equations with more variables than equations
48. Rank of a homogenous system of linear equations
49. Rank of a matrix
50. Basis of F^n
51. Matrix of linear transformation
52. Coordinatization over a basis
53. Basis changer
54. Basis change is an isomorphic linear transformation
55. Vector spaces are isomorphic iff their dimensions are same
56. Canonical decomposition of a linear transformation
57. Eigenvalues and Eigenvectors