Orthonormal Matrix

An Orthonormal Matrix, [math]\displaystyle{ Q_t }[/math] is an orthogonal matrix (with mutually orthogonal unit vectors) that is equal to its inverse, that is [math]\displaystyle{ Q^\mathrm{T}=Q^{-1} }[/math] .

• Context:
  • It can be obtained for a set of linearly independent column vectors [math]\displaystyle{ \{v_1,v_2,\dots, v_n\} }[/math] by Gram-Schmidt process.
  • It must have eigenvalues which are real or complex conjugate pairs and of magnitude 1.
  • It must have determinant values -1 or +1.
  • It can be used in Matrix Decomposition Processes like QR Decomposition, Singular Value Decomposition, Eigen Decomposition, and Polar Decomposition.
• Example(s):
  • an Identity Matrix.
  • a Rotation Matrix, [math]\displaystyle{ A=\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta\end{bmatrix} }[/math].
  • [math]\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{5}} \end{bmatrix} }[/math], because each column vector is of magnitude 1, [math]\displaystyle{ \left( \begin{vmatrix}\begin{bmatrix}\frac{1}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \end{bmatrix}\end{vmatrix} = 1 = \begin{vmatrix}\begin{bmatrix}\frac{2}{\sqrt{5}} \\ -\frac{1}{\sqrt{5}} \end{bmatrix}\end{vmatrix} \right) }[/math]; and the inner product of the column vectors are 0, [math]\displaystyle{ \left( \begin{bmatrix}\frac{1}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} \end{bmatrix} . \begin{bmatrix} \frac{2}{\sqrt{5}} \\ -\frac{1}{\sqrt{5}} \end{bmatrix}=0 \right) }[/math].
  • an x-Axis Reflection Matrix.
  • a Coordinate Axes Permutation Matrix.
  • …
• Counter-Example(s):
  • a Non-Normalized Orthogonal Matrix, such as [math]\displaystyle{ \begin{bmatrix}1 & 3 \\ 2 & 4 \end{bmatrix} }[/math].
• See: QR Decomposition, Gaussian Matrix.

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References

2015

• https://en.wikibooks.org/wiki/Linear_Algebra/Topic:_Orthonormal_Matrices
  • QUOTE: … Those matrices have that when the columns are written as vectors then they are of length one and are mutually orthogonal. Such a matrix is called an orthonormal matrix or orthogonal matrix (the first term is commonly used to mean not just that the columns are orthogonal, but also that they have length one).

2013

• http://en.wikipedia.org/wiki/Orthogonal_matrix
  • In linear algebra, an orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors).

    Equivalently, a matrix Q is orthogonal if its transpose is equal to its inverse: :[math]\displaystyle{ Q^\mathrm{T}=Q^{-1}, \, }[/math] which entails :[math]\displaystyle{ Q^\mathrm{T} Q = Q Q^\mathrm{T} = I, \, }[/math] where I is the identity matrix.

    An orthogonal matrix Q is necessarily invertible (with inverse Q⁻¹ = Q^T), unitary (Q⁻¹ = Q*), and normal (Q*Q = QQ*). The determinant of any orthogonal matrix is either +1 or −1. As a linear transformation, an orthogonal matrix preserves the dot product of vectors, and therefore acts as an isometry of Euclidean space, such as a rotation or reflection. In other words, it is a unitary transformation.

    The set of n × n orthogonal matrices forms a group O(n), known as the orthogonal group. The subgroup SO(n) consisting of orthogonal matrices with determinant +1 is called the special orthogonal group, and each of its elements is a special orthogonal matrix. As a linear transformation, every special orthogonal matrix acts as a rotation.

    The complex analogue of an orthogonal matrix is a unitary matrix.

• http://en.wikipedia.org/wiki/Orthogonal_matrix#Examples
  • Below are a few examples of small orthogonal matrices and possible interpretations.
    • [math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad (\text{identity transformation}) }[/math]
    • An instance of a 2×2 rotation matrix: [math]\displaystyle{ R(16.26^\circ) = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{bmatrix} = \begin{bmatrix} 0.96 & -0.28 \\ 0.28 & \;\;\,0.96 \\ \end{bmatrix} \qquad (\text{rotation by }16.26^\circ ) }[/math]
    • [math]\displaystyle{ \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix} \qquad (\text{reflection across }x\text{-axis}) }[/math]
    • [math]\displaystyle{ \begin{bmatrix} 0 & -0.80 & -0.60 \\ 0.80 & -0.36 & \;\;\,0.48 \\ 0.60 & \;\;\,0.48 & -0.64 \end{bmatrix} \qquad \left( \begin{align}&\text{rotoinversion:} \\&\text{axis }(0,-3/5,4/5),\text{ angle }90^{\circ}\end{align}\right) }[/math]

      ***[math]\displaystyle{ \begin{bmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{bmatrix} \qquad (\text{permutation of coordinate axes}) }[/math]

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