Symmetric operator iff hermitian

Dependencies:

  1. Conjugate Transpose and Hermitian
  2. Basis of a vector space
  3. Gram-Schmidt Process
  4. Symmetric operator
  5. Canonical decomposition of a linear transformation
  6. Matrix of linear transformation
  7. Matrices form an inner-product space
  8. Inner product is anti-linear in second argument
  9. Transpose of product

Let $B = [v_1, v_2, \ldots, v_n]$ be an orthonormal basis of $V$, where $V$ is an inner product space of finite dimension $n$ on the field $F$.

Let $L: V \mapsto V$ be a linear transformation. By the canonical decomposition theorem, $L = RTR^{-1}$, where $R: F^n \mapsto V$ and $R([a_1, a_2, \ldots, a_n]) = \sum_{i=1}^n a_iv_i$ and $R$ is an isomorphic linear transformation.

Since every linear transformation from $F^n$ to $F^n$ can be expressed as matrix pre-multiplication, $T(x) = Ax$.

Then $L$ is symmetric iff $A = A^*$ ($A^*$ is the conjugate transpose of $A$).

Conversely, let $A$ be a square matrix. Then $T(u) = Au$ is a symmetric linear operator iff $A = A^*$.

Proof

Define $\langle x, y \rangle = y^*x$ to be the inner product on $F^n$.

Lemma 1: $\langle R(a), R(b) \rangle = \langle a, b \rangle$

Let $a = [a_1, a_2, \ldots, a_n]$ and $b = [b_1, b_2, \ldots, b_n]$.

\begin{align} \langle R(a), R(b) \rangle &= \left\langle \sum_{i=1}^n a_iv_i, \sum_{i=1}^n b_iv_i \right\rangle \\ &= \sum_{i=1}^n \sum_{j=1}^n a_i \overline{b_j} \langle v_i, v_j \rangle \tag{by (anti-)linearity} \\ &= \sum_{i=1}^n a_i \overline{b_i} \tag{$B$ is orthonormal} \\ &= b^*a = \langle a, b \rangle \end{align}

Lemma 2

Let $R(x) = u$ and $R(y) = v$.

\begin{align} & \langle L(u), v \rangle \\ &= \langle R(T(R^{-1}(u))), v \rangle \\ &= \langle R(T(x)), R(y) \rangle \\ &= \langle T(x), y \rangle \tag{by lemma 1} \\ &= \langle Ax, y \rangle \\ &= y^*Ax \end{align}

\begin{align} & \langle u, L(v) \rangle \\ &= \langle u, R(T(R^{-1}(v))) \rangle \\ &= \langle R(x), R(T(y)) \rangle \\ &= \langle x, T(y) \rangle \tag{by lemma 1} \\ &= \langle x, Ay \rangle \\ &= (Ay)^*x = y^*A^*x \end{align}

Lemma 3

Suppose $\forall x, y \in F^n, y^*Ax = 0$.

\[ y^*Ax = \sum_{i=1}^n \sum_{j=1}^n (y^*)[1, i] A[i, j] x[j, 1] = \sum_{i=1}^n \sum_{j=1}^n \overline{y_i} x_j A[i, j] \]

Plugging in $x = e_j$ and $y = e_i$ in the above equation ($e_k$ is a column vector with all entries 0 except the $k^{\textrm{th}}$ entry, which is 1), we get $y^*Ax = A[i, j]$.

Therefore, $(\forall x, y \in F^n, y^*Ax = 0) \iff A = 0$.

Conclusion

\begin{align} & \forall u, v \in V, \langle L(u), v \rangle = \langle u, L(v) \rangle \\ &\iff \forall x, y \in F^n, y^*Ax = y^*A^*x \tag{by lemma 2 and $\because R$ is a bijection} \\ &\iff \forall x, y \in F^n, y^*(A^*-A)x = 0 \\ &\iff A = A^* \tag{by lemma 3} \end{align}

Converse

Let $A$ be an $n$ by $n$ matrix. Then $T(u) = Au$ is a linear transformation.

\[ \langle T(u), v \rangle - \langle u, T(v) \rangle = \langle Au, v \rangle - \langle u, Av \rangle = v^*(Au) - (Av)^*u = v^*Au - v^*A^*u = v^*(A - A^*)u \]

\begin{align} & T \textrm{ is symmetric} \\ &\iff \forall u, v \in F^n, \langle T(u), v \rangle = \langle u, T(v) \rangle \\ &\iff \forall u, v \in F^n, v^*(A - A^*)u = 0 \\ &\iff A = A^* \tag{by lemma 3} \end{align}

Dependency for:

  1. Orthogonally diagonalizable iff hermitian

Info:

Transitive dependencies:

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