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