Orthogonal Matrix

An Orthogonal Matrix is a real square matrix whose columns and rows are orthogonal vectors.

• Context:
  • It can range from being an Orthonormal Matrix to being a Non-Normalized Orthogonal Matrix.
  • …
  • It is used in matrix decomposition processes like QR Decomposition, Singular Value Decomposition, Eigen Decomposition, Polar Decomposition, etc.
• Example(s):
  • an Identity Matrix.
  • a Rotation Matrix.
  • [math]\displaystyle{ \begin{bmatrix}x & 0 & 0\\ 0 & y & 0 \\0 & 0 & z\end{bmatrix} }[/math].
  • [math]\displaystyle{ \begin{bmatrix}1 & 3 \\ 2 & 4 \end{bmatrix} }[/math], along with its normalized version [math]\displaystyle{ \begin{bmatrix}\frac{1}{\sqrt{5}} & \frac{2}{\sqrt{5}} \\ \frac{2}{\sqrt{5}} & -\frac{1}{\sqrt{5}} \end{bmatrix} }[/math].
  • an x-Axis Reflection Matrix.
  • a Coordinate Axes Permutation Matrix.
  • …
• Counter-Example(s):
  • a Non-Square Unitary Matrix.
  • a Strictly Triangular Matrix.
• See: Unitary Matrix, Orthogonal Transformation, Unit Vector, Orthonormality, Invertible Matrix, Normal Matrix.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/orthogonal_matrix Retrieved:2015-2-16.
  • 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), i.e. :[math]\displaystyle{ Q^\mathrm{T} Q = Q Q^\mathrm{T} = I, }[/math]

    where I is the identity matrix.

    This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse: :[math]\displaystyle{ Q^\mathrm{T}=Q^{-1}, \, }[/math]

    An orthogonal matrix Q is necessarily invertible (with inverse ), unitary () and therefore normal () in the reals. 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.

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