Orthogonal matrix
Two vectors are orthogonal when their dot product equals zero. If the lenght of each vector is 1 then the vectors are called orthonormal because they are both orthogonal and normalized.
ν ⋅ w = 0
or
ν ⋅ wT = 0
which is intuitive considering that on line is orthogonal with another if it is perpendicular to it. An orthogonal is a type of square matrix whose columns and rows are orthonormal unit vectors, e.g. perpendicular and have a lenght or magnitude of 1.
An orthogonal matrix is a square matrix whose rows are mutually orthonormal and whose columns are mutually orthonormal. An orthogonal matrix is defined formally as follows
QT ⋅ Q = Q ⋅ QT = I
Where Q is the orthogonal matrix, QT indicates the transpose of Q, and I is the identity matrix. A matrix is orthogonal if its transpose is equal to its inverse.
QT = Q−1
The determinants of orthogonal matrices are equal to either +1 or −1 due to QT Q = In :
det(QTQ) = det(QT) det(Q) = (det Q)2 = det In = 1 or det Q = ±1.
The orthogonal n × n matrices form a subgroup of GLn(ℝ) denoted by (On(ℝ), ⋅) and called the orthogonal group:
On = { A ∈ GLn(ℝ)| ATA = I}
Obviously, the group O(n) is a subset of the group GL(n, R) of all real n × n invertible matrices. Since the group operations in GL(n, R) and O(n) are the same (the multiplication of n × n matrices), it follows that O(n) is a subgroup of GL(n, R). On the other hand, the set of all orthogonal n × n matrices with the determinant det A = +1 forms a subgroup SOn (special orthogonal matrices) of O(n): this set is closed under matrix multiplication since det(AB) = det A · det B = 1 · 1 = 1, as well as under inversion det(A −1 ) = (detA)−1 = 1.
SOn = { A ∈ GLn(ℝ)| ATA = I, det A = 1}
The groups SOn, n = 2, 3, ..., play important roles in applications in mathematics, as well as in physics; for instance, the rotational symmetry of atoms [SO3 commutes with the energy operator] gives as a result the electron orbits in atoms.