What is the condition for orthogonal matrix?
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In linear algebra, an orthogonal matrix or real orthogonal matrix is a square matrix with realentries whose columns and rows are orthogonal unit vectors (i.e., orthonormalvectors), i.e.
{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}
where {\displaystyle I} is the identity matrix.
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
{\displaystyle Q^{\mathrm {T} }=Q^{-1}.}
An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary(Q−1 = Q∗) and therefore normal (Q∗Q = QQ∗) 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.
{\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,}
where {\displaystyle I} is the identity matrix.
This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse:
{\displaystyle Q^{\mathrm {T} }=Q^{-1}.}
An orthogonal matrix Q is necessarily invertible (with inverse Q−1 = QT), unitary(Q−1 = Q∗) and therefore normal (Q∗Q = QQ∗) 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.
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