Orthogonally diagonalizable iff hermitian

Dependencies:

1. Conjugate Transpose and Hermitian
2. Orthogonal matrix
3. Transpose of product
4. Symmetric operator iff hermitian
5. Symmetric operator on V has a basis of orthonormal eigenvectors
6. All eigenvalues of a symmetric operator are real
7. Real matrix with real eigenvalues has real eigenvectors
8. Diagonalization

Let $A$ be an $n$ by $n$ matrix over $\mathbb{C}$.

$A$ is said to be orthogonally diagonalizable iff $A = PDP^*$, where $P$ is an orthogonal matrix and $D$ is a diagonal matrix with real entries.

We'll prove 2 things:

• $A$ is orthogonally diagonalizable iff $A = A^*$.
• Let $A = A^*$. Let $[v_1, v_2, \ldots, v_n]$ be eigenvectors of $A$ with $[\lambda_1, \lambda_2, \ldots, \lambda_n]$ as the corresponding eigenvalues. Let $P$ be a matrix where the $i^{\textrm{th}}$ column is $v_i$. Then $P$ is orthogonal and $A = PDP^*$. Also, if $A$ is real, then $P$ is real.

Proof of 'only-if' part

Let $A$ be orthogonally diagonalizable.

\[ A = PDP^* \implies A^* = (PDP^*)^* = PD^*P^* = PDP^* = A \]

Proof of 'if' part

Let $A = A^*$. This means that the operator $T(u) = Au$ is a symmetric operator over the vector space $\mathbb{C}^n$.

A symmetric operator on a finite-dimensional vector space $V$ over field $\mathbb{C}$ has $\dim(V)$ orthonormal eigenvectors. Therefore, $A$ has $n$ orthonormal eigenvectors. Let $[v_1, v_2, \ldots, v_n]$ be the eigenvectors and $[\lambda_1, \lambda_2, \ldots, \lambda_n]$ be the corresponding eigenvalues. $\forall i, \lambda_i \in \mathbb{R}$, since all eigenvalues of a symmetric operator are real. If $A$ is real, $[v_1, v_2, \ldots, v_n]$ are real.

Let $P$ be the matrix whose columns are $[v_1, v_2, \ldots, v_n]$. Then $P$ is orthogonal and $P$ is real if $A$ is real. Let $D$ be a diagonal matrix where $D[i, i] = \lambda_i$. Therefore, $AP = PD$, which implies that $A = APP^* = PDP^*$. Therefore, $A$ is orthgonally diagonalizable.

Dependency for:

1. Bounding matrix quadratic form using eigenvalues

Info:

• Depth: 16
• Number of transitive dependencies: 92

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. Reduced Row Echelon Form (RREF)
41. Conjugate Transpose and Hermitian
42. Transpose of product
43. Conjugation of matrices is homomorphic
44. Submatrix
45. Determinant
46. Determinant of upper triangular matrix
47. Swapping last 2 rows of a matrix negates its determinant
48. Trace of a matrix
49. Matrices over a field form a vector space
50. Row space
51. Matrices form an inner-product space
52. Elementary row operation
53. Determinant after elementary row operation
54. Every elementary row operation has a unique inverse
55. Row equivalence of matrices
56. Row equivalent matrices have the same row space
57. RREF is unique
58. Identity matrix
59. Inverse of a matrix
60. Inverse of product
61. Elementary row operation is matrix pre-multiplication
62. Row equivalence matrix
63. Equations with row equivalent matrices have the same solution set
64. Rank of a homogenous system of linear equations
65. Rank of a matrix
66. Basis of a vector space
67. Linearly independent set is not bigger than a span
68. Homogeneous linear equations with more variables than equations
69. A set of dim(V) linearly independent vectors is a basis
70. Basis of F^n
71. Matrix of linear transformation
72. Coordinatization over a basis
73. Basis changer
74. Basis change is an isomorphic linear transformation
75. Vector spaces are isomorphic iff their dimensions are same
76. Canonical decomposition of a linear transformation
77. Eigenvalues and Eigenvectors
78. All eigenvalues of a symmetric operator are real
79. Real matrix with real eigenvalues has real eigenvectors
80. Diagonalization
81. Symmetric operator iff hermitian
82. Linearly independent set can be expanded into a basis
83. Full-rank square matrix in RREF is the identity matrix
84. A matrix is full-rank iff its determinant is non-0
85. Characteristic polynomial of a matrix
86. Degree and monicness of a characteristic polynomial
87. Full-rank square matrix is invertible
88. AB = I implies BA = I
89. Determinant of product is product of determinants
90. Every complex matrix has an eigenvalue
91. Symmetric operator on V has a basis of orthonormal eigenvectors
92. Orthogonal matrix