Positive definite iff eigenvalues are positive

Dependencies:

1. Positive definite
2. Eigenvalues and Eigenvectors
3. All eigenvalues of a hermitian matrix are real
4. Bounding matrix quadratic form using eigenvalues

Let $A$ be a real $n$ by $n$ symmetric matrix. This means that all its eigenvalues are real. Let $\lambda_{\textrm{min}}$ and $\lambda_{\textrm{max}}$ be the minimum and maximum eigenvalues. Then

• $A$ is positive definite iff $\lambda_{\textrm{min}} > 0$.
• $A$ is positive semidefinite iff $\lambda_{\textrm{min}} \ge 0$.
• $A$ is negative definite iff $\lambda_{\textrm{max}} < 0$.
• $A$ is negative semidefinite iff $\lambda_{\textrm{max}} \le 0$.

Proof

\[ \forall u \in \mathbb{R}^n, \lambda_{\textrm{min}}u^Tu \le u^TAu \le \lambda_{\textrm{max}}u^Tu \] Also, the above bounds are tight. The theorems of this page directly follow from the above bound.

Dependency for:

1. Bound on eigenvalues of sum of matrices

Info:

• Depth: 18
• Number of transitive dependencies: 96

Transitive dependencies:

1. /linear-algebra/vector-spaces/condition-for-subspace
2. /linear-algebra/matrices/gauss-jordan-algo
3. /complex-numbers/conjugate-product-abs
4. /complex-numbers/conjugation-is-homomorphic
5. /complex-numbers/complex-numbers
6. /linear-algebra/eigenvectors/cayley-hamilton-theorem
7. /misc/fundamental-theorem-of-algebra
8. /sets-and-relations/composition-of-bijections-is-a-bijection
9. /sets-and-relations/equivalence-relation
10. Group
11. Ring
12. Polynomial
13. Vector
14. Dot-product of vectors
15. Integral Domain
16. Comparing coefficients of a polynomial with disjoint variables
17. 0x = 0 = x0
18. Field
19. Vector Space
20. Linear independence
21. Span
22. Incrementing a linearly independent set
23. Linear transformation
24. Composition of linear transformations
25. Vector space isomorphism is an equivalence relation
26. Inner product space
27. Inner product is anti-linear in second argument
28. Orthogonality and orthonormality
29. Gram-Schmidt Process
30. A set of mutually orthogonal vectors is linearly independent
31. Symmetric operator
32. A field is an integral domain
33. Semiring
34. Matrix
35. Stacking
36. System of linear equations
37. Product of stacked matrices
38. Transpose of stacked matrix
39. Matrix multiplication is associative
40. Positive definite
41. Reduced Row Echelon Form (RREF)
42. Conjugate Transpose and Hermitian
43. Transpose of product
44. Conjugation of matrices is homomorphic
45. Submatrix
46. Determinant
47. Determinant of upper triangular matrix
48. Swapping last 2 rows of a matrix negates its determinant
49. Trace of a matrix
50. Matrices over a field form a vector space
51. Row space
52. Matrices form an inner-product space
53. Elementary row operation
54. Determinant after elementary row operation
55. Every elementary row operation has a unique inverse
56. Row equivalence of matrices
57. Row equivalent matrices have the same row space
58. RREF is unique
59. Identity matrix
60. Inverse of a matrix
61. Inverse of product
62. Elementary row operation is matrix pre-multiplication
63. Row equivalence matrix
64. Equations with row equivalent matrices have the same solution set
65. Basis of a vector space
66. Linearly independent set is not bigger than a span
67. Homogeneous linear equations with more variables than equations
68. Rank of a homogenous system of linear equations
69. Rank of a matrix
70. A set of dim(V) linearly independent vectors is a basis
71. Basis of F^n
72. Matrix of linear transformation
73. Coordinatization over a basis
74. Basis changer
75. Basis change is an isomorphic linear transformation
76. Vector spaces are isomorphic iff their dimensions are same
77. Canonical decomposition of a linear transformation
78. Eigenvalues and Eigenvectors
79. All eigenvalues of a hermitian matrix are real
80. All eigenvalues of a symmetric operator are real
81. Real matrix with real eigenvalues has real eigenvectors
82. Diagonalization
83. Symmetric operator iff hermitian
84. Linearly independent set can be expanded into a basis
85. Full-rank square matrix in RREF is the identity matrix
86. A matrix is full-rank iff its determinant is non-0
87. Characteristic polynomial of a matrix
88. Degree and monicness of a characteristic polynomial
89. Full-rank square matrix is invertible
90. AB = I implies BA = I
91. Determinant of product is product of determinants
92. Every complex matrix has an eigenvalue
93. Symmetric operator on V has a basis of orthonormal eigenvectors
94. Orthogonal matrix
95. Orthogonally diagonalizable iff hermitian
96. Bounding matrix quadratic form using eigenvalues