Matrix of orthonormal basis change

Dependencies:

Let $P$ and $Q$ be orthonormal bases of vector space $F^n$, where $F$ is a field. Inner-product of vectors is defined to be $\langle u, v \rangle = v^*u$. Let $T$ be a basis-change function from $P$ to $Q$.

Since basis-change is a linear transformation and every linear transformation on finite-dimensional vector spaces can be expressed as matrix pre-multiplication, $T$ has an associated matrix $A$ such that $T(u) = Au$ ($u$ is treated as a column vector).

We have to prove that $A$ is orthogonal.

Proof

Let $E = [e_1, e_2, \ldots, e_n]$ be the standard basis of $F^n$. Let $P = [p_1, p_2, \ldots, p_n]$ and $Q = [q_1, q_2, \ldots, q_n]$.

Let $T_P$ be the basis change function from $E$ to $P$. Let $T_Q$ be the basis change function from $E$ to $Q$.

\[ \left(T_QT_P^{-1}\right)\left(\sum_{i=1}^n a_ip_i \right) = T_Q\left(T_P^{-1}\left(\sum_{i=1}^n a_ip_i \right)\right) = T_Q\left(\sum_{i=1}^n a_ie_i\right) = \sum_{i=1}^n a_iq_i \]

Therefore, $T_QT_P^{-1}$ is a basis change function from $P$ to $Q$.

Let $B$ be the matrix whose columns are vectors of $P$. Let $C$ be the matrix whose columns are vectors of $Q$. Since $P$ has orthonormal vectors, $B$ is an orthogonal matrix. Since $Q$ has orthonormal vectors, $C$ is an orthogonal matrix.

Let $a = [a_1, a_2, \ldots, a_n]$. \begin{align} & T_P(a)_j \\ &= \left(\sum_{i=1}^n a_i p_i \right)_j \\ &= \sum_{i=1}^n a_i (p_i)_j \\ &= \sum_{i=1}^n B[j, i] a_i \\ &= (Ba)_j \end{align}

Therefore, $T_P(a) = Ba$. Similarly, $T_Q(a) = Ca$.

Let $d = T_P(a)$. \[ d = T_P(a) = Ba = B T_P^{-1}(d) \Rightarrow T_P^{-1}(d) = B^{-1}d = B^*d \]

\[ T(u) = T_QT_P^{-1}(u) = CB^*u \] Therefore, the matrix of $T$ is $CB^*$.

\[ (CB^*)^*(CB^*) = BC^*CB^* = BB^* = I \] Therefore, $CB^*$ is orthogonal. Therefore, the matrix of $T$ is orthogonal.

Dependency for: None

Info:

Depth: 12
Number of transitive dependencies: 58

Transitive dependencies:

/linear-algebra/matrices/gauss-jordan-algo
/linear-algebra/vector-spaces/condition-for-subspace
/complex-numbers/conjugate-product-abs
/complex-numbers/conjugation-is-homomorphic
/sets-and-relations/equivalence-relation
Group
Ring
Polynomial
Vector
Dot-product of vectors
Field
Vector Space
Inner product space
Orthogonality and orthonormality
Linear independence
Span
Linear transformation
Integral Domain
Comparing coefficients of a polynomial with disjoint variables
Semiring
Matrix
Stacking
System of linear equations
Transpose of stacked matrix
Product of stacked matrices
Matrix multiplication is associative
Trace of a matrix
Conjugation of matrices is homomorphic
Transpose of product
Reduced Row Echelon Form (RREF)
Elementary row operation
Every elementary row operation has a unique inverse
Row equivalence of matrices
Matrices over a field form a vector space
Row space
Row equivalent matrices have the same row space
RREF is unique
Matrices form an inner-product space
Identity matrix
Full-rank square matrix in RREF is the identity matrix
Inverse of a matrix
Inverse of product
Elementary row operation is matrix pre-multiplication
Row equivalence matrix
Equations with row equivalent matrices have the same solution set
Basis of a vector space
Linearly independent set is not bigger than a span
Homogeneous linear equations with more variables than equations
Rank of a homogenous system of linear equations
Rank of a matrix
Basis of F^n
Matrix of linear transformation
Coordinatization over a basis
Basis changer
Basis change is an isomorphic linear transformation
Full-rank square matrix is invertible
AB = I implies BA = I
Orthogonal matrix