Orthonormal basis change matrix

Dependencies:

  1. Inner product space
  2. Basis of a vector space
  3. Orthogonality and orthonormality
  4. Orthogonal matrix

Let $U = [u_1, u_2, \ldots, u_m]$ be an orthonormal basis for an inner-product space $\mathcal{S}$ over the field $F$. Let $V = [v_1, v_2, \ldots, v_m]$ be another orthonormal basis for $\mathcal{S}$.

Since $U$ is a basis, every vector in $\mathcal{S}$, including those in $V$, can be represented as a linear combination of $U$. Therefore, let $A$ be an $m$-by-$m$ matrix on $F$ such that \[ \forall i, v_i = \sum_{j=1}^m A[i, j] u_j \] Such an $A$ is called a basis change matrix from $V$ to $U$ (because pre-multiplying the coordinates of a vector w.r.t. $V$ by $A^T$ would give the coordinates w.r.t. $U$).

I'll prove that $A$ is orthogonal.

Proof

\begin{align} I[i, j] &= \langle v_i, v_j \rangle \tag{$\because$ $V$ is orthonormal} \\ &= \left\langle \sum_{p=1}^m A[i, p] u_p , \sum_{q=1}^m A[j, q] u_q \right\rangle \\ &= \sum_{p=1}^m \sum_{q=1}^m A[i, p] A[j, q] \langle u_p, u_q \rangle \tag{linearity of $\mathcal{S}$} \\ &= \sum_{p=1}^m \sum_{q=1}^m A[i, p] I[p, q] A[j, q] \tag{$\because$ $U$ is orthonormal} \\ &= (AIA^T)[i, j] = (AA^T)[i, j] \end{align}

Therefore, $AA^T = I$. Since $A$ is a square matrix, $A$ is orthogonal.

Dependency for:

  1. Standard normal random vector on vector space

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. Group
  7. Ring
  8. Polynomial
  9. Vector
  10. Dot-product of vectors
  11. Field
  12. Vector Space
  13. Inner product space
  14. Orthogonality and orthonormality
  15. Linear independence
  16. Span
  17. Integral Domain
  18. Comparing coefficients of a polynomial with disjoint variables
  19. Semiring
  20. Matrix
  21. Stacking
  22. System of linear equations
  23. Transpose of stacked matrix
  24. Product of stacked matrices
  25. Matrix multiplication is associative
  26. Trace of a matrix
  27. Transpose of product
  28. Reduced Row Echelon Form (RREF)
  29. Elementary row operation
  30. Every elementary row operation has a unique inverse
  31. Row equivalence of matrices
  32. Matrices over a field form a vector space
  33. Row space
  34. Row equivalent matrices have the same row space
  35. RREF is unique
  36. Matrices form an inner-product space
  37. Identity matrix
  38. Full-rank square matrix in RREF is the identity matrix
  39. Inverse of a matrix
  40. Inverse of product
  41. Elementary row operation is matrix pre-multiplication
  42. Row equivalence matrix
  43. Equations with row equivalent matrices have the same solution set
  44. Basis of a vector space
  45. Linearly independent set is not bigger than a span
  46. Homogeneous linear equations with more variables than equations
  47. Rank of a homogenous system of linear equations
  48. Rank of a matrix
  49. Full-rank square matrix is invertible
  50. AB = I implies BA = I
  51. Orthogonal matrix