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/vector-spaces/condition-for-subspace
/linear-algebra/matrices/gauss-jordan-algo
/complex-numbers/conjugate-product-abs
/complex-numbers/conjugation-is-homomorphic
/sets-and-relations/equivalence-relation
Group
Ring
Polynomial
Vector
Dot-product of vectors
Integral Domain
Comparing coefficients of a polynomial with disjoint variables
Field
Vector Space
Linear independence
Span
Linear transformation
Inner product space
Orthogonality and orthonormality
Semiring
Matrix
Stacking
System of linear equations
Product of stacked matrices
Transpose of stacked matrix
Matrix multiplication is associative
Reduced Row Echelon Form (RREF)
Transpose of product
Conjugation of matrices is homomorphic
Trace of a matrix
Matrices over a field form a vector space
Row space
Matrices form an inner-product space
Elementary row operation
Every elementary row operation has a unique inverse
Row equivalence of matrices
Row equivalent matrices have the same row space
RREF is unique
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 in RREF is the identity matrix
Full-rank square matrix is invertible
AB = I implies BA = I
Orthogonal matrix