📂Matrix Algebra

Properties of Orthogonal Matrices

Properties¹

An orthogonal matrix has the following properties:

(a) The transpose of an orthogonal matrix is also an orthogonal matrix.

(b) The inverse of an orthogonal matrix is an orthogonal matrix.

(c) The product of two orthogonal matrices is an orthogonal matrix.

(d) The determinant of an orthogonal matrix is either $1$ or $-1$.

$$ \det(A)=\pm 1 $$

Proof

(a)

Let’s say $A$ is an orthogonal matrix. Let’s say $B$ is the transpose of $A$.

$$ B=A^{T} $$

Then, the following equation holds:

$$ B^{-1} = (A^{T})^{-1} = (A^{-1})^{-1} = A = B^{T} $$

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(b)

Let’s say $A$ is an orthogonal matrix. Let’s say $B$ is the inverse of $A$.

$$ B = A^{-1} $$

Then, since $A$ is an orthogonal matrix, and $(A^{-1})^{T} = (A^{T})^{-1}$ holds, the following equation holds:

$$ B^{-1} = (A^{-1})^{-1} = (A^{T})^{-1} = (A^{-1})^{T} = B^{T} $$

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(c)

Let’s say $A$, $B$ are orthogonal matrices of size $n \times n$. Then, it suffices to show that $(AB) (AB)^{T}$. Since $(AB)^{T}=B^{T}A^{T}$ holds, the following equation holds:

$$ \begin{align*} (AB)(AB)^{T} &= (AB) (B^{T}A^{T}) \\ &= (AB)(B^{-1}A^{-1}) \\ &= AA^{-1} \\ &= I \end{align*} $$

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(d)

Let’s say $A$ is an orthogonal matrix. Then, since the determinant of the product is equal to the product of the determinants, we obtain the following equation:

$$ \begin{align*} \det(I) &= \det(AA^{T}) \\ &= \det(A) \det(A^{T}) \end{align*} $$

Also, since the determinant of the transpose is equal to the transpose of the determinant, we obtain the following equation:

$$ 1 = \det(I) = \left( \det(A) \right)^2 $$

Therefore,

$$ \det(A) = \pm 1 $$