Semi-orthogonal matrix
In linear algebra, a semi-orthogonal matrix is a rectangular matrix with real entries whose columns (or rows, depending on the dimensions) form an orthonormal set, satisfying either A^T A = I_n for an m \times n matrix with m \geq n or A A^T = I_m for m < n.[1] This property ensures that the matrix acts as a partial isometry, preserving the Euclidean norm of vectors in the appropriate subspace.[2] Such matrices are fundamental in matrix decompositions, including the QR decomposition, where a full-rank m \times n matrix A (with m \geq n) can be factored as A = QR, with Q semi-orthogonal and R upper triangular.[1] They also appear in the polar decomposition of rectangular matrices, expressing A = PV where P is semi-orthogonal and V is positive semidefinite.[1] For symmetric matrices, semi-orthogonal matrices facilitate spectral decompositions by providing orthonormal bases for eigenspaces of reduced dimension.[1] Beyond decompositions, semi-orthogonal matrices find applications in numerical linear algebra and statistics, such as in least squares estimation and principal component analysis, where they enable efficient orthogonal projections.[1] In signal processing, they model phenomena like turbulent fluctuations by maintaining orthogonality in subspaces.[2] Recent research explores additional constraints, such as when semi-orthogonal matrices have row vectors of equal lengths, which occurs under specific scaling conditions and relates to Grassmannian coordinates for column spaces.[3] These properties distinguish semi-orthogonal matrices from fully orthogonal square matrices, which satisfy both A^T A = I and A A^T = I.[1]Definition and Terminology
Formal Definition
A semi-orthogonal matrix is a rectangular real matrix Q \in \mathbb{R}^{m \times n} whose entries satisfy specific orthonormality conditions depending on the dimensions m and n.[4] When m \geq n (a tall matrix), the columns of Q form an orthonormal set, satisfying Q^\top Q = I_n, where I_n denotes the n \times n identity matrix; in this case, Q preserves the Euclidean norm for all vectors x \in \mathbb{R}^n, i.e., \| Q x \|_2 = \| x \|_2.[4] When m \leq n (a short matrix), the rows of Q form an orthonormal set, satisfying Q Q^\top = I_m, where I_m denotes the m \times m identity matrix; here, Q preserves the Euclidean norm for all vectors x in the row space of Q, i.e., \| Q x \|_2 = \| x \|_2 for x in the row space of Q.[4] The prefix "semi-" highlights the partial orthogonality arising from the non-square shape, in contrast to a full orthogonal matrix, which is square and satisfies both Q^\top Q = Q Q^\top = I (or the analogous condition for unitary matrices over the complex numbers).[4]Equivalent Characterizations
A semi-orthogonal matrix can be characterized as a partial isometry, meaning it maps vectors in the orthogonal complement of its kernel to vectors of the same Euclidean norm in the codomain. Specifically, for a matrix Q \in \mathbb{R}^{m \times n}, \| Qx \|_2 = \| x \|_2 holds for all x such that x \perp \ker(Q).[5] This property extends the notion of an isometry to rectangular matrices, where the isometric action is restricted to a subspace of the domain.[5] In the tall case where m \geq n and Q^\top Q = I_n, Q serves as a sub-isometry by preserving the Euclidean norm for all vectors in the entire domain \mathbb{R}^n, as the kernel is trivial under full column rank. In the short case where m \leq n and Q Q^\top = I_m, Q preserves norms on its row space, which is the orthogonal complement of the kernel. These cases highlight how semi-orthogonal matrices embed isometric mappings into higher- or lower-dimensional spaces.[5] An equivalent matrix-theoretic characterization involves the singular values: for a semi-orthogonal matrix Q, the singular values satisfy \sigma_i(Q) = 1 for i = 1, \dots, \min(m,n), with any remaining singular values being zero if the matrix is not square. This follows from the fact that the non-zero singular values correspond to the unit eigenvalues of Q^\top Q or Q Q^\top.[5] Unlike full isometries, which are square orthogonal matrices preserving norms across the entire domain and codomain, semi-orthogonal matrices act as isometries only on appropriate subspaces, allowing for dimension mismatch while maintaining partial norm preservation.[5]Core Properties
Orthonormality Conditions
A semi-orthogonal matrix satisfies specific orthonormality conditions that distinguish it from fully orthogonal matrices, applying to either its columns or rows depending on the matrix dimensions. For a tall semi-orthogonal matrix Q \in \mathbb{R}^{m \times n} with m > n, the columns \mathbf{q}_i and \mathbf{q}_j (for i, j = 1, \dots, n) are orthonormal, meaning their inner products satisfy \mathbf{q}_i^\top \mathbf{q}_j = \delta_{ij}, where \delta_{ij} is the Kronecker delta (equal to 1 if i = j and 0 otherwise). This condition ensures that the columns form an orthonormal basis for the column space of Q. Similarly, for a short semi-orthogonal matrix Q \in \mathbb{R}^{m \times n} with m < n, the rows \mathbf{r}_i and \mathbf{r}_j (for i, j = 1, \dots, m) satisfy \mathbf{r}_i \mathbf{r}_j^\top = \delta_{ij}, establishing orthonormality among the rows. These pairwise orthonormality requirements extend to a preservation of inner products within the appropriate subspace. For the tall case, the transformation Q preserves the Euclidean inner product for vectors \mathbf{x}, \mathbf{y} \in \mathbb{R}^n, such that \langle Q \mathbf{x}, Q \mathbf{y} \rangle = \langle \mathbf{x}, \mathbf{y} \rangle. In the short case, this preservation holds for vectors in the row space, aligning with the orthonormality of the rows. As a direct consequence, the relevant Gram matrix equals the identity on the corresponding dimensions: Q^\top Q = I_n for tall matrices and Q Q^\top = I_m for short matrices. This property underscores the role of semi-orthogonal matrices in forming idempotent projections, where Q Q^\top (for tall) or Q^\top Q (for short) acts as an orthogonal projector onto the column or row space, respectively.Norm Preservation
A key property of semi-orthogonal matrices is their preservation of the Euclidean norm on specific subspaces, distinguishing them as partial isometries in linear transformations. For a tall semi-orthogonal matrix [Q](/page/Q) \in \mathbb{R}^{m \times n} with m > n and satisfying Q^\top Q = I_n, the Euclidean norm is preserved under the mapping [Q](/page/Q): \mathbb{R}^n \to \mathbb{R}^m, meaning \| [Q](/page/Q) x \|_2 = \| x \|_2 for all x \in \mathbb{R}^n.[6] This norm preservation arises directly from the orthonormality condition on the columns of Q. A brief sketch of the derivation shows that \| Q x \|_2^2 = x^\top Q^\top Q x = x^\top I_n x = x^\top x = \| x \|_2^2, confirming the equality of norms without altering lengths. For a short semi-orthogonal matrix Q \in \mathbb{R}^{m \times n} with m < n and satisfying Q Q^\top = I_m, the preservation holds on the row space of Q, so \| Q x \|_2 = \| x \|_2 for all x in the row space of Q. Geometrically, the columns of a tall Q (or rows of a short Q) form an orthonormal frame for the subspace, ensuring that vectors are embedded or projected without distortion in length, akin to a rigid rotation or reflection within that frame. In contrast to general matrices, which typically distort the Euclidean norm through scaling, shearing, or contraction/expansion, semi-orthogonal matrices induce rigid transformations that maintain distances on their defined subspaces.Full Rank and Singular Values
A semi-orthogonal matrix Q \in \mathbb{R}^{m \times n} has full rank equal to \min(m, n), as its columns (or rows, depending on the orientation) form an orthonormal set and are thus linearly independent.[7] The singular values of Q consist of exactly \min(m, n) values equal to 1 and the remaining |m - n| values equal to 0, reflecting its structure as a partial isometry.[8] For the case where m \geq n (tall matrix), the eigenvalues of Q^\top Q are all 1 (with multiplicity n), confirming the non-zero singular values are 1.[9] Similarly, for m < n (short matrix), the eigenvalues of Q Q^\top are all 1 (with multiplicity m). This singular value structure implies that a tall semi-orthogonal matrix Q (with m \geq n) admits a left inverse given by Q^\top, since Q^\top Q = I_n.[10] Conversely, a short semi-orthogonal matrix admits a right inverse Q^\top, as Q Q^\top = I_m.[10] In contrast to fully orthogonal square matrices, where all singular values are exactly 1, semi-orthogonal matrices exhibit a partial spectrum of 1's due to their rectangular nature.Examples
Tall Matrices
A tall semi-orthogonal matrix features orthonormal columns, providing an isometric embedding from \mathbb{R}^n into \mathbb{R}^m for m > n. A basic example is the $2 \times 1 matrix Q = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \end{pmatrix}, whose single column is a unit vector, satisfying Q^\top Q = 1.[11] Another example arises in the Householder QR decomposition process, where the thin Q factor is a tall matrix with orthonormal columns; for instance, consider the $3 \times 2 matrix Q = \begin{pmatrix} \frac{1}{\sqrt{3}} & 0 \\ \frac{1}{\sqrt{3}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} \end{pmatrix}, with columns that are unit vectors and mutually orthogonal, as Q^\top Q = I_2.[11][12] Geometrically, the columns of this Q span a plane in \mathbb{R}^3, and left-multiplication by Q maps vectors from \mathbb{R}^2 into \mathbb{R}^3 without distorting lengths or angles within that plane.[13] To illustrate norm preservation, for the first example with input x = 1, we have Qx = Q and \|Qx\|_2 = 1 = \|x\|_2; similarly, for the second example with x = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, Qx is the first column of Q with \|Qx\|_2 = 1 = \|x\|_2, and for x = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, Qx is the second column with matching norms.[11]Short Matrices
A short semi-orthogonal matrix is an m \times n matrix with m < n whose rows form an orthonormal set in \mathbb{R}^n.[14] A basic example is the $1 \times 2 matrixQ = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}.
The single row has Euclidean norm \sqrt{ \left( \frac{1}{\sqrt{2}} \right)^2 + \left( \frac{1}{\sqrt{2}} \right)^2 } = \sqrt{ \frac{1}{2} + \frac{1}{2} } = 1, so Q Q^T = 1, verifying row-orthonormality.[14] For a $2 \times 3 example from coordinate averaging, consider rows derived by normalizing averages of coordinate directions, such as the first row from averaging the first and second standard basis vectors in \mathbb{R}^3:
\mathbf{r}_1 = \frac{1}{\sqrt{2}} ( \mathbf{e}_1 + \mathbf{e}_2 ) = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \end{pmatrix},
and the second row orthogonal to it and unit length, obtained via Gram-Schmidt on an average involving the third coordinate, such as
\mathbf{r}_2 = \frac{1}{\sqrt{2}} ( \mathbf{e}_1 - \mathbf{e}_2 ) = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \end{pmatrix}.
The matrix is
Q = \begin{pmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 \end{pmatrix}.
Each row has norm 1, and their dot product is \frac{1}{2} - \frac{1}{2} + 0 = 0, so Q Q^T = I_2.[14] This spans the xy-plane subspace. Such matrices enable isometric projections of \mathbb{R}^3 onto \mathbb{R}^2 within the row space, embedding the parameter space isometrically via Q^T.[14] To check norm preservation on row space vectors, take y = \begin{pmatrix} 1 \\ 0 \end{pmatrix} \in \mathbb{R}^2. Then Q^T y = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} \\ 0 \end{pmatrix} \in \mathbb{R}^3, with \| Q^T y \| = 1 = \| y \|. Applying Q recovers Q (Q^T y) = y, preserving the norm. A similar check holds for y = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, yielding Q^T y = \begin{pmatrix} \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} \\ 0 \end{pmatrix} with norm 1.[14]