Products, Inverses, and Powers of Orthogonal Matrices

Orthogonality Survives Products and Inverses

Orthogonal matrices are useful not only because they preserve lengths and inner products, but also because their structure is retained under basic operations. If two orthogonal matrices can be composed, the product is orthogonal; reversing an orthogonal transformation also gives an orthogonal matrix. These facts let us calculate repeated transformations by studying powers of a single matrix, rather than checking each power from scratch.

Throughout this tutorial, matrices are real and square. A matrix \(\mathbf{Q}\in\mathbb{R}^{n\times n}\) is orthogonal when \(\mathbf{Q}^{\top}\mathbf{Q}=\mathbf{I}_n\). This condition says that the columns of \(\mathbf{Q}\) are orthonormal. For a square matrix, it also gives \(\mathbf{Q}\mathbf{Q}^{\top}=\mathbf{I}_n\) and \(\mathbf{Q}^{-1}=\mathbf{Q}^{\top}\): by the Invertible Matrix Theorem, a square matrix with a left inverse is invertible, and its left inverse must be its inverse.

Proof. For the product, use the transpose rule \((\mathbf{A}\mathbf{B})^{\top}=\mathbf{B}^{\top}\mathbf{A}^{\top}\), associativity of multiplication, and the orthogonality identities:

Thus \(\mathbf{A}\mathbf{B}\) is orthogonal by definition. For the inverse, \(\mathbf{A}^{-1}=\mathbf{A}^{\top}\), so

Therefore \(\mathbf{A}^{-1}\) is orthogonal as well. \(\square\)

The order of factors matters in matrix multiplication, but the proof does not require the matrices to commute: the middle factors \(\mathbf{A}^{\top}\mathbf{A}\) are adjacent and equal to \(\mathbf{I}_n\). The same attention to order is needed for inverses. In particular, \((\mathbf{A}\mathbf{B})^{-1}=\mathbf{B}^{-1}\mathbf{A}^{-1}\), not \(\mathbf{A}^{-1}\mathbf{B}^{-1}\). To check the order, multiply \(\mathbf{A}\mathbf{B}\) by \(\mathbf{B}^{-1}\mathbf{A}^{-1}\) on the right: the result is \(\mathbf{A}\mathbf{I}_n\mathbf{A}^{-1}=\mathbf{I}_n\).

Integer Powers of an Orthogonal Matrix

For a square matrix \(\mathbf{Q}\), define \(\mathbf{Q}^0=\mathbf{I}_n\), and for a positive integer \(k\), define \(\mathbf{Q}^k\) as the product of \(k\) copies of \(\mathbf{Q}\). For a negative integer \(-k\), where \(k>0\) and \(\mathbf{Q}\) is invertible, define \(\mathbf{Q}^{-k}=(\mathbf{Q}^{-1})^k\). Every orthogonal matrix is invertible, so all of its integer powers are defined.

Proof. First consider nonnegative integers. The identity \(\mathbf{Q}^0=\mathbf{I}_n\) is orthogonal, because \(\mathbf{I}_n^{\top}\mathbf{I}_n=\mathbf{I}_n\). Suppose \(\mathbf{Q}^k\) is orthogonal for some \(k\geq0\). By the product theorem, \(\mathbf{Q}^k\mathbf{Q}\) is orthogonal. Associativity and the definition of powers give \(\mathbf{Q}^k\mathbf{Q}=\mathbf{Q}^{k+1}\). Induction proves orthogonality for every \(k\geq0\). For \(k<0\), write \(k=-j\) with \(j>0\). The inverse theorem shows that \(\mathbf{Q}^{-1}\) is orthogonal, so the nonnegative case applied to \(\mathbf{Q}^{-1}\) shows that \((\mathbf{Q}^{-1})^j=\mathbf{Q}^{-j}\) is orthogonal. This covers every integer. \(\square\)

The proof explains why a power need not be checked by multiplying its transpose by itself each time. Once the original matrix is known to be orthogonal, closure under products and inverses does the work. This is a structural conclusion; a particular power can still be computed explicitly when the entries or the geometry are of interest.

Powers of Rotation Matrices

For a real angle \(\theta\), write the standard planar rotation matrix as

The columns are orthonormal because \(\cos^2\theta+\sin^2\theta=1\) and their dot product is \(-\cos\theta\sin\theta+\sin\theta\cos\theta=0\). Thus \(\mathbf{R}(\theta)\) is orthogonal. Multiplication combines angles: the entries of the product are given by the angle-addition identities, so \(\mathbf{R}(\alpha)\mathbf{R}(\beta)=\mathbf{R}(\alpha+\beta)\). In particular, repeated multiplication yields \(\mathbf{R}(\theta)^k=\mathbf{R}(k\theta)\) for nonnegative integers \(k\). For negative integers the same formula follows because \(\mathbf{R}(\theta)^{-1}=\mathbf{R}(-\theta)\).

Powers of Reflection Matrices

A reflection across a line through the origin making angle \(\theta\) with the positive horizontal axis has matrix

This matrix is symmetric. Its square is the identity: the diagonal entries of the square are \(\cos^2(2\theta)+\sin^2(2\theta)=1\), and both off-diagonal entries are \(\cos(2\theta)\sin(2\theta)-\sin(2\theta)\cos(2\theta)=0\). Therefore \(\mathbf{S}(\theta)^2=\mathbf{I}_2\), which also proves it is orthogonal, since \(\mathbf{S}(\theta)^{\top}\mathbf{S}(\theta)=\mathbf{I}_2\). It follows that even powers equal \(\mathbf{I}_2\), while odd powers equal \(\mathbf{S}(\theta)\): each successive pair of factors can be replaced by \(\mathbf{I}_2\). The same parity rule holds for negative integer powers because the inverse of this matrix is itself.

Common Pitfall: Reversing Factors or Assuming Powers Simplify

A frequent error is to reverse factors incorrectly when taking an inverse. For the matrices in the first example, \(\mathbf{A}^{-1}=\begin{bmatrix}0&1\\-1&0\end{bmatrix}\) and \(\mathbf{B}^{-1}=\mathbf{B}\). The incorrect order gives

which is not \((\mathbf{A}\mathbf{B})^{-1}=\begin{bmatrix}0&1\\1&0\end{bmatrix}\). The correction is to reverse both the factors and their order: \((\mathbf{A}\mathbf{B})^{-1}=\mathbf{B}^{-1}\mathbf{A}^{-1}\). Multiplying the corrected expression gives the actual inverse and verifies the rule.

Another error is to treat every orthogonal matrix like a reflection and conclude that its square is the identity. The rotation in the previous example contradicts this: \(\mathbf{R}^2=\begin{bmatrix}-\tfrac12&-\tfrac{\sqrt3}{2}\\\tfrac{\sqrt3}{2}&-\tfrac12\end{bmatrix}\ne\mathbf{I}_2\). Orthogonality guarantees that powers remain orthogonal; it does not say those powers equal the identity. The special parity pattern for reflection powers follows from the additional fact \(\mathbf{S}^2=\mathbf{I}_2\).

These closure properties are valuable whenever a transformation is applied repeatedly or when several length-preserving transformations are composed. They provide a quick structural check: a product of orthogonal matrices must be orthogonal, and an inverse must be orthogonal. For actual entries, use the relevant structure as well—angle addition for the rotation matrices above, and the square-equals-identity relation for the displayed reflections.