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In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix
rotates points in the xyCartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation using a rotation matrix R, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv.
Since matrix multiplication has no effect on the zero vector (the coordinates of the origin), rotation matrices can only be used to describe rotations about the origin of the coordinate system. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in geometry, physics, and computer graphics.
Rotation matrices are square matrices, with real entries. More specifically they can be characterized as orthogonal matrices with determinant 1:
In some literature, the term rotation is generalized to include improper rotations, characterized by orthogonal matrices with determinant −1 (instead of +1). These combine proper rotations with reflections (which invert orientation). In other cases, where reflections are not being considered, the label proper may be dropped. This convention is followed in this article.
The set of all orthogonal matrices of size n with determinant +1 forms a group known as the special orthogonal group SO(n). The set of all orthogonal matrices of size n with determinant +1 or 1 forms the (general) orthogonal group O(n).
In two dimensions every rotation matrix has the following form:
This rotates column vectors by means of the following matrix multiplication:
So the coordinates (x',y') of the point (x,y) after rotation are:^{[1]}
The direction of vector rotation is counterclockwise if θ is positive (e.g. 90°), and clockwise if θ is negative (e.g. 90°). Thus the clockwise rotation matrix is found as:
Note that the twodimensional case is the only nontrivial (e.g. one dimension) case where the rotation matrices group is commutative, so that it does not matter the order in which multiple rotations are performed.
If a standard righthanded Cartesian coordinate system is used, with the x axis to the right and the y axis up, the rotation R(θ) is counterclockwise. If a lefthanded Cartesian coordinate system is used, with x directed to the right but y directed down, R(θ) is clockwise. Such nonstandard orientations are rarely used in mathematics but are common in 2D computer graphics, which often have the origin in the top left corner and the yaxis down the screen or page.^{[2]}
See below for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.
Particularly useful are the matrices for 90° and 180° rotations:
A basic rotation (also called elemental rotation) is a rotation about one of the axes of a Coordinate system. The following three basic rotation matrices rotate vectors by an angle θ about the x, y, or z axis, in three dimensions:
For column vectors, each of these basic vector rotations appears counterclockwise when the axis about which they occur points toward the observer, the coordinate system is righthanded, and the angle θ is positive. R_{z}, for instance, would rotate toward the yaxis a vector aligned with the xaxis, as can easily be checked by operating with R_{z} on the vector (1,0,0):
This is similar to the rotation produced by the above mentioned 2D rotation matrix. See below for alternative conventions which may apparently or actually invert the sense of the rotation produced by these matrices.
Other rotation matrices can be obtained from these three using matrix multiplication. For example, the product
represents a rotation whose yaw, pitch, and roll angles are α, β, and γ, respectively. More formally, it is an extrinsic rotation whose Euler angles are α, β, γ, about axes z, y, x respectively. Similarly, the product
represents an extrinsic rotation whose Euler angles are α, β, γ about axes y, x, z.
These matrices produce the desired effect only if they are used to premultiply column vectors (see Ambiguities for more details).
It has been suggested that this section be merged into Rotation formalisms in three dimensions#Rotation matrix ↔ Euler axis/angle. (Discuss) Proposed since September 2013. 
Every rotation in three dimensions is defined by its axis — a direction that is left fixed by the rotation — and its angle — the amount of rotation about that axis (Euler rotation theorem).
There are several methods to compute an axis and an angle from a rotation matrix (see also axisangle). Here, we only describe the method based on the computation of the eigenvectors and eigenvalues of the rotation matrix. It is also possible to use the trace of the rotation matrix.
Given a 3x3 rotation matrix R, a vector u parallel to the rotation axis must satisfy
since the rotation of around the rotation axis must result in . The equation above may be solved for which is unique up to a scalar factor.
Further, the equation may be rewritten
which shows that is the null space of .
Viewed another way, is an eigenvector of R corresponding to the eigenvalue (every rotation matrix must have this eigenvalue).
To find the angle of a rotation, once the axis of the rotation is known, select a vector perpendicular to the axis. Then the angle of the rotation is the angle between and .
A much easier method, however, is to calculate the trace (i.e. the sum of the diagonal elements of the rotation matrix) which is . Care should be taken to select the right sign for the angle to match the chosen axis.
For some applications, it is helpful to be able to make a rotation with a given axis. Given a unit vector u = (u_{x}, u_{y}, u_{z}), where u_{x}^{2} + u_{y}^{2} + u_{z}^{2} = 1, the matrix for a rotation by an angle of θ about an axis in the direction of u is
This can be written more concisely as
where is the cross product matrix of u, is the tensor product and I is the Identity matrix. This is a matrix form of Rodrigues' rotation formula, with
If the 3D space is righthanded, this rotation will be counterclockwise for an observer placed so that the axis u goes in her direction (Righthand rule).
In three dimensions, for any rotation matrix acting on , where a is a rotation axis and θ a rotation angle,
Some of these properties can be generalised to any number of dimensions. In other words, they hold for any rotation matrix .
For instance, in two dimensions the properties hold with the following exceptions:


In Euclidean geometry, a rotation is an example of an isometry, a transformation that moves points without changing the distances between them. Rotations are distinguished from other isometries by two additional properties: they leave (at least) one point fixed, and they leave "handedness" unchanged. By contrast, a translation moves every point, a reflection exchanges left and righthanded ordering, and a glide reflection does both.
A rotation that does not leave "handedness" unchanged is an improper rotation or a rotoinversion.
If we take the fixed point as the origin of a Cartesian coordinate system, then every point can be given coordinates as a displacement from the origin. Thus we may work with the vector space of displacements instead of the points themselves. Now suppose (p_{1},…,p_{n}) are the coordinates of the vector p from the origin, O, to point P. Choose an orthonormal basis for our coordinates; then the squared distance to P, by Pythagoras, is
which we can compute using the matrix multiplication
A geometric rotation transforms lines to lines, and preserves ratios of distances between points. From these properties we can show that a rotation is a linear transformation of the vectors, and thus can be written in matrix form, Qp. The fact that a rotation preserves, not just ratios, but distances themselves, we can state as
or
Because this equation holds for all vectors, p, we conclude that every rotation matrix, Q, satisfies the orthogonality condition,
Rotations preserve handedness because they cannot change the ordering of the axes, which implies the special matrix condition,
Equally important, we can show that any matrix satisfying these two conditions acts as a rotation.
The inverse of a rotation matrix is its transpose, which is also a rotation matrix:
The product of two rotation matrices is a rotation matrix:
For n greater than 2, multiplication of n×n rotation matrices is not commutative.
Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n×n rotation matrices form a group, which for n > 2 is nonabelian. Called a special orthogonal group, and denoted by SO(n), SO(n,R), SO_{n}, or SO_{n}(R), the group of n×n rotation matrices is isomorphic to the group of rotations in an ndimensional space. This means that multiplication of rotation matrices corresponds to composition of rotations, applied in lefttoright order of their corresponding matrices.
The interpretation of a rotation matrix can be subject to many ambiguities.
In most cases the effect of the ambiguity is equivalent to the effect of a transposition of the rotation matrix.
Consider the 3×3 rotation matrix
If Q acts in a certain direction, v, purely as a scaling by a factor λ, then we have
so that
Thus λ is a root of the characteristic polynomial for Q,
Two features are noteworthy. First, one of the roots (or eigenvalues) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the axis of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is 2 cos θ (the negated linear term). This factorization is of interest for 3×3 rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any n×n rotation matrix. If the dimension, n, is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree n and thus n eigenvalues. And since a rotation matrix commutes with its transpose, it is a normal matrix, so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of twodimensional subspaces, at most ^{n}⁄_{2} of them.
The sum of the entries on the main diagonal of a matrix is called the trace; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for 2×2 and 3×3 rotation matrices that the trace reveals the angle of rotation, θ, in the twodimensional (sub)space. For a 2×2 matrix the trace is 2 cos(θ), and for a 3×3 matrix it is 1+2 cos(θ). In the threedimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any 3×3 rotation matrix a rotation axis and an angle, and these completely determine the rotation.
The constraints on a 2×2 rotation matrix imply that it must have the form
with a^{2}+b^{2} = 1. Therefore we may set a = cos θ and b = sin θ, for some angle θ. To solve for θ it is not enough to look at a alone or b alone; we must consider both together to place the angle in the correct quadrant, using a twoargument arctangent function.
Now consider the first column of a 3×3 rotation matrix,
Although a^{2}+b^{2} will probably not equal 1, but some value r^{2} < 1, we can use a slight variation of the previous computation to find a socalled Givens rotation that transforms the column to
zeroing b. This acts on the subspace spanned by the x and y axes. We can then repeat the process for the xz subspace to zero c. Acting on the full matrix, these two rotations produce the schematic form
Shifting attention to the second column, a Givens rotation of the yz subspace can now zero the z value. This brings the full matrix to the form
which is an identity matrix. Thus we have decomposed Q as
An n×n rotation matrix will have (n−1)+(n−2)+⋯+2+1, or
entries below the diagonal to zero. We can zero them by extending the same idea of stepping through the columns with a series of rotations in a fixed sequence of planes. We conclude that the set of n×n rotation matrices, each of which has n^{2} entries, can be parameterized by n(n−1)/2 angles.
xzx_{w}  xzy_{w}  xyx_{w}  xyz_{w} 
yxy_{w}  yxz_{w}  yzy_{w}  yzx_{w} 
zyz_{w}  zyx_{w}  zxz_{w}  zxy_{w} 
xzx_{b}  yzx_{b}  xyx_{b}  zyx_{b} 
yxy_{b}  zxy_{b}  yzy_{b}  xzy_{b} 
zyz_{b}  xyz_{b}  zxz_{b}  yxz_{b} 
In three dimensions this restates in matrix form an observation made by Euler, so mathematicians call the ordered sequence of three angles Euler angles. However, the situation is somewhat more complicated than we have so far indicated. Despite the small dimension, we actually have considerable freedom in the sequence of axis pairs we use; and we also have some freedom in the choice of angles. Thus we find many different conventions employed when threedimensional rotations are parameterized for physics, or medicine, or chemistry, or other disciplines. When we include the option of world axes or body axes, 24 different sequences are possible. And while some disciplines call any sequence Euler angles, others give different names (Euler, Cardano, TaitBryan, Rollpitchyaw) to different sequences.
One reason for the large number of options is that, as noted previously, rotations in three dimensions (and higher) do not commute. If we reverse a given sequence of rotations, we get a different outcome. This also implies that we cannot compose two rotations by adding their corresponding angles. Thus Euler angles are not vectors, despite a similarity in appearance as a triple of numbers.
A 3×3 rotation matrix like
suggests a 2×2 rotation matrix,
is embedded in the upper left corner:
This is no illusion; not just one, but many, copies of ndimensional rotations are found within (n+1)dimensional rotations, as subgroups. Each embedding leaves one direction fixed, which in the case of 3×3 matrices is the rotation axis. For example, we have
fixing the x axis, the y axis, and the z axis, respectively. The rotation axis need not be a coordinate axis; if u = (x,y,z) is a unit vector in the desired direction, then
where c_{θ} = cos θ, s_{θ} = sin θ, is a rotation by angle θ leaving axis u fixed.
A direction in (n+1)dimensional space will be a unit magnitude vector, which we may consider a point on a generalized sphere, S^{n}. Thus it is natural to describe the rotation group SO(n+1) as combining SO(n) and S^{n}. A suitable formalism is the fiber bundle,
where for every direction in the "base space", S^{n}, the "fiber" over it in the "total space", SO(n+1), is a copy of the "fiber space", SO(n), namely the rotations that keep that direction fixed.
Thus we can build an n×n rotation matrix by starting with a 2×2 matrix, aiming its fixed axis on S^{2} (the ordinary sphere in threedimensional space), aiming the resulting rotation on S^{3}, and so on up through S^{n−1}. A point on S^{n} can be selected using n numbers, so we again have n(n−1)/2 numbers to describe any n×n rotation matrix.
In fact, we can view the sequential angle decomposition, discussed previously, as reversing this process. The composition of n−1 Givens rotations brings the first column (and row) to (1,0,…,0), so that the remainder of the matrix is a rotation matrix of dimension one less, embedded so as to leave (1,0,…,0) fixed.
When an n×n rotation matrix, Q, does not include −1 as an eigenvalue, so that none of the planar rotations of which it is composed are 180° rotations, then Q+I is an invertible matrix. Most rotation matrices fit this description, and for them we can show that (Q−I)(Q+I)^{−1} is a skewsymmetric matrix, A. Thus A^{T} = −A; and since the diagonal is necessarily zero, and since the upper triangle determines the lower one, A contains n(n−1)/2 independent numbers. Conveniently, I−A is invertible whenever A is skewsymmetric; thus we can recover the original matrix using the Cayley transform,
which maps any skewsymmetric matrix A to a rotation matrix. In fact, aside from the noted exceptions, we can produce any rotation matrix in this way. Although in practical applications we can hardly afford to ignore 180° rotations, the Cayley transform is still a potentially useful tool, giving a parameterization of most rotation matrices without trigonometric functions.
In three dimensions, for example, we have (Cayley 1846)
If we condense the skew entries into a vector, (x,y,z), then we produce a 90° rotation around the x axis for (1,0,0), around the y axis for (0,1,0), and around the z axis for (0,0,1). The 180° rotations are just out of reach; for, in the limit as x goes to infinity, (x,0,0) does approach a 180° rotation around the x axis, and similarly for other directions.
For the 2D case, a rotation matrix can be decomposed into three shear matrices (Paeth 1986):
This is useful, for instance, in computer graphics, since shears can be implemented with fewer multiplication instructions than rotating a bitmap directly. On modern computers, this may not matter, but it can be relevant for very old or lowend microprocessors.
We have established that n×n rotation matrices form a group, the special orthogonal group, SO(n). This algebraic structure is coupled with a topological structure, in that the operations of multiplication and taking the inverse (which here is merely transposition) are continuous functions of the matrix entries. Thus SO(n) is a classic example of a topological group. (In purely topological terms, it is a compact manifold.) Furthermore, the operations are not only continuous, but smooth, so SO(n) is a differentiable manifold and a Lie group.^{[4]}
Most properties of rotation matrices depend very little on the dimension, n; yet in Lie group theory we see systematic differences between even dimensions and odd dimensions. As well, there are some irregularities below n = 5; for example, SO(4) is, anomalously, not a simple Lie group, but instead isomorphic to the product of S^{3} and SO(3).
Associated with every Lie group is a Lie algebra, a linear space equipped with a bilinear alternating product called a bracket. The algebra for SO(n) is denoted by
and consists of all skewsymmetric n×n matrices (as implied by differentiating the orthogonality condition, I = Q^{T}Q). The bracket, [A_{1},A_{2}], of two skewsymmetric matrices is defined to be A_{1}A_{2}−A_{2}A_{1}, which is again a skewsymmetric matrix. This Lie algebra bracket captures the essence of the Lie group product via infinitesimals.
For 2×2 rotation matrices, the Lie algebra so(2) is a onedimensional vector space, mere multiples of
Here the bracket always vanishes, which tells us that, in two dimensions, rotations commute. Not so in any higher dimension.
For 3×3 rotation matrices, one has a threedimensional vector space with the convenient basis
The Lie brackets of these generators are as follows,
We can conveniently identify any matrix in this Lie algebra with a vector in R^{3},
Under this identification, the so(3) bracket has a memorable description; it is the vector cross product,
The matrix identified with a vector v is also memorable, because
Notice this implies that v is in the null space of the skewsymmetric matrix with which it is identified, because v×v is always the zero vector.
Connecting the Lie algebra to the Lie group is the exponential map, which is defined using the standard matrix exponential series for e^{A},^{[5]}
For any skewsymmetric matrix A, exp(A) is always a rotation matrix. Note that this exponential map of skewsymmetric matrices to rotation matrices is quite different from the Cayley transform discussed earlier, differing to 3rd order,
An important practical example is the 3×3 case, where we have seen we can identify every skewsymmetric matrix with a vector ω = θ u, where u = (x,y,z) is a unit magnitude vector. Recall that u is in the null space of the matrix associated with ω; so that, if we use a basis with u as the z axis, the final column and row will be zero. Thus, we know in advance that the exponential matrix must leave u fixed. It is mathematically impossible to supply a straightforward formula for such a basis as a function of u (its existence would violate the hairy ball theorem); but direct exponentiation is possible, and yields
where c = cos ^{θ}⁄_{2}, s = sin ^{θ}⁄_{2}. We recognize this as our matrix for a rotation around axis u by the angle θ.
In any dimension, if we choose some nonzero A and consider all its scalar multiples, exponentiation yields rotation matrices along a geodesic of the group manifold, forming a oneparameter subgroup of the Lie group. More broadly, the exponential map provides a homeomorphism between a neighborhood of the origin in the Lie algebra and a neighborhood of the identity in the Lie group. In fact, we can produce any rotation matrix as the exponential of some skewsymmetric matrix, so for these groups the exponential map is a surjection.
Suppose we are given A and B in the Lie algebra. Their exponentials, exp(A) and exp(B), are rotation matrices, which we can multiply. Since the exponential map is a surjection, we know that, for some C in the Lie algebra, exp(A)exp(B) = exp(C), and so we may write
When exp(A) and exp(B) commute, then C = A+B, mimicking the behavior of complex exponentiation. However, the general case is given by the more elaborate BCH formula, a series expansion of nested brackets.^{[6]} For matrices, the bracket is the same operation as the commutator, which monitors lack of commutativity in multiplication. This general expansion unfolds as follows,
Representation of a rotation matrix as a sequential angle decomposition, as in Euler angles, may tempt one to treat rotations as a vector space, but the higher order terms in the BCH formula deprecate such an approach for large angles.
We again take special interest in the 3×3 case, where [A,B] equals the cross product, A×B. If A and B are linearly independent, then A, B, and A×B provide a complete basis; if not, then A and B commute. Evidently, in this dimension, the infinite expansion in the BCH formula for group conposition has a compact form, as C = αA+βB+γA×B for suitable coefficients.^{[7]} (See the straightforward 2×2 derivation for SU(2).)
The Lie group of n×n rotation matrices, SO(n), is a compact and pathconnected manifold, and thus locally compact and connected. However, it is not simply connected, so Lie theory tells us it is a kind of "shadow" (a homomorphic image) of a universal covering group. Often the covering group, which in this case is the spin group denoted by Spin(n), is simpler and more natural to work with.^{[8]}
In the case of planar rotations, SO(2) is topologically a circle, S^{1}. Its universal covering group, Spin(2), is isomorphic to the real line, R, under addition. In other words, whenever we use angles of arbitrary magnitude, which we often do, we are essentially taking advantage of the convenience of the "mother space". Every 2×2 rotation matrix is produced by a countable infinity of angles, separated by integer multiples of 2π. Correspondingly, the fundamental group of SO(2) is isomorphic to the integers, Z.
In the case of spatial rotations, SO(3) is topologically equivalent to threedimensional real projective space, RP^{3}. Its universal covering group, Spin(3), is isomorphic to the 3sphere, S^{3}. Every 3×3 rotation matrix is produced by two opposite points on the sphere. Correspondingly, the fundamental group of SO(3) is isomorphic to the twoelement group, Z_{2}. We can also describe Spin(3) as isomorphic to quaternions of unit norm under multiplication, or to certain 4×4 real matrices, or to 2×2 complex special unitary matrices.
Concretely, a unit quaternion, q, with
produces the rotation matrix
This is our third version of this matrix, here as a rotation around the now nonunit axis vector (x,y,z) by angle 2θ, where cos θ = w and sin θ = (x,y,z). (The proper sign for sin θ is implied once the signs of the axis components are fixed.)
Many features of this case are the same for higher dimensions. The coverings are all twotoone, with SO(n), n > 2, having fundamental group Z_{2}. The natural setting for these groups is within a Clifford algebra. And the action of the rotations is produced by a kind of "sandwich", denoted by qvq^{∗}.
The matrices in the Lie algebra are not themselves rotations; the skewsymmetric matrices are derivatives, proportional differences of rotations. An actual "differential rotation", or infinitesimal rotation matrix has the form
where dθ is vanishingly small.
These matrices do not satisfy all the same properties as ordinary finite rotation matrices under the usual treatment of infinitesimals .^{[9]} To understand what this means, consider
First, test the orthogonality condition, Q^{T}Q = I. The product is
differing from an identity matrix by second order infinitesimals, discarded here. So, to first order, an infinitesimal rotation matrix is an orthogonal matrix.
Next, examine the square of the matrix,
Again discarding second order effects, note that the angle simply doubles. This hints at the most essential difference in behavior, which we can exhibit with the assistance of a second infinitesimal rotation,
Compare the products dA_{x}dA_{y} to dA_{y}dA_{x},
Since dθ dφ is second order, we discard it: thus, to first order, multiplication of infinitesimal rotation matrices is commutative. In fact,
again to first order. In other words, the order in which infinitesimal rotations are applied is irrelevant.
This useful fact makes, for example, derivation of rigid body rotation relatively simple. But we must always be careful to distinguish (the first order treatment of) these infinitesimal rotation matrices from both finite rotation matrices and from derivatives of rotation matrices (namely skewsymmetric matrices). Contrast the behavior of finite rotation matrices in the BCH formula above with that of infinitesimal rotation matrices, where all the commutator terms will be second order infinitesimals so we do have a bona fide vector space. (Technically, this dismissal of any second order terms amounts to Group contraction.)
It has been suggested that this section be merged into Rotation formalisms in three dimensions#Conversion formulae between formalisms. (Discuss) Proposed since September 2013. 
We have seen the existence of several decompositions that apply in any dimension, namely independent planes, sequential angles, and nested dimensions. In all these cases we can either decompose a matrix or construct one. We have also given special attention to 3×3 rotation matrices, and these warrant further attention, in both directions (Stuelpnagel 1964).
Given the unit quaternion q = (w,x,y,z), the equivalent lefthanded (PostMultiplied) 3×3 rotation matrix is
Now every quaternion component appears multiplied by two in a term of degree two, and if all such terms are zero what's left is an identity matrix. This leads to an efficient, robust conversion from any quaternion – whether unit, nonunit, or even zero – to a 3×3 rotation matrix.
Nq = w^2 + x^2 + y^2 + z^2 if Nq > 0.0 then s = 2/Nq else s = 0.0 X = x*s; Y = y*s; Z = z*s wX = w*X; wY = w*Y; wZ = w*Z xX = x*X; xY = x*Y; xZ = x*Z yY = y*Y; yZ = y*Z; zZ = z*Z [ 1.0(yY+zZ) xYwZ xZ+wY ] [ xY+wZ 1.0(xX+zZ) yZwX ] [ xZwY yZ+wX 1.0(xX+yY) ]
Freed from the demand for a unit quaternion, we find that nonzero quaternions act as homogeneous coordinates for 3×3 rotation matrices. The Cayley transform, discussed earlier, is obtained by scaling the quaternion so that its w component is 1. For a 180° rotation around any axis, w will be zero, which explains the Cayley limitation.
The sum of the entries along the main diagonal (the trace), plus one, equals 4−4(x^{2}+y^{2}+z^{2}), which is 4w^{2}. Thus we can write the trace itself as 2w^{2}+2w^{2}−1; and from the previous version of the matrix we see that the diagonal entries themselves have the same form: 2x^{2}+2w^{2}−1, 2y^{2}+2w^{2}−1, and 2z^{2}+2w^{2}−1. So we can easily compare the magnitudes of all four quaternion components using the matrix diagonal. We can, in fact, obtain all four magnitudes using sums and square roots, and choose consistent signs using the skewsymmetric part of the offdiagonal entries.
t = Q_{xx}+Q_{yy}+Q_{zz} (trace of Q) r = sqrt(1+t) w = 0.5*r x = copysign(0.5*sqrt(1+Q_{xx}Q_{yy}Q_{zz}), Q_{zy}Q_{yz}) y = copysign(0.5*sqrt(1Q_{xx}+Q_{yy}Q_{zz}), Q_{xz}Q_{zx}) z = copysign(0.5*sqrt(1Q_{xx}Q_{yy}+Q_{zz}), Q_{yx}Q_{xy})
where copysign(x,y) is x with the sign of y:
Alternatively, use a single square root and division
t = Q_{xx}+Q_{yy}+Q_{zz} r = sqrt(1+t) s = 0.5/r w = 0.5*r x = (Q_{zy}Q_{yz})*s y = (Q_{xz}Q_{zx})*s z = (Q_{yx}Q_{xy})*s
This is numerically stable so long as the trace, t, is not negative; otherwise, we risk dividing by (nearly) zero. In that case, suppose Q_{xx} is the largest diagonal entry, so x will have the largest magnitude (the other cases are similar); then the following is safe.
t = Q_{xx}+Q_{yy}+Q_{zz} r = sqrt(1+Q_{xx}Q_{yy}Q_{zz}) s = 0.5/r w = (Q_{zy}Q_{yz})*s x = 0.5*r y = (Q_{xy}+Q_{yx})*s z = (Q_{zx}+Q_{xz})*s
If the matrix contains significant error, such as accumulated numerical error, we may construct a symmetric 4×4 matrix,
and find the eigenvector, (w,x,y,z), of its largest magnitude eigenvalue. (If Q is truly a rotation matrix, that value will be 1.) The quaternion so obtained will correspond to the rotation matrix closest to the given matrix^{[dubious – discuss]} (BarItzhack 2000).
If the n×n matrix M is nonsingular, its columns are linearly independent vectors; thus the Gram–Schmidt process can adjust them to be an orthonormal basis. Stated in terms of numerical linear algebra, we convert M to an orthogonal matrix, Q, using QR decomposition. However, we often prefer a Q "closest" to M, which this method does not accomplish. For that, the tool we want is the polar decomposition (Fan & Hoffman 1955; Higham 1989).
To measure closeness, we may use any matrix norm invariant under orthogonal transformations. A convenient choice is the Frobenius norm, Q−M_{F}, squared, which is the sum of the squares of the element differences. Writing this in terms of the trace, Tr, our goal is,
Though written in matrix terms, the objective function is just a quadratic polynomial. We can minimize it in the usual way, by finding where its derivative is zero. For a 3×3 matrix, the orthogonality constraint implies six scalar equalities that the entries of Q must satisfy. To incorporate the constraint(s), we may employ a standard technique, Lagrange multipliers, assembled as a symmetric matrix, Y. Thus our method is:
Consider a 2×2 example. Including constraints, we seek to minimize
Taking the derivative with respect to Q_{xx}, Q_{xy}, Q_{yx}, Q_{yy} in turn, we assemble a matrix.
In general, we obtain the equation
so that
where Q is orthogonal and S is symmetric. To ensure a minimum, the Y matrix (and hence S) must be positive definite. Linear algebra calls QS the polar decomposition of M, with S the positive square root of S^{2} = M^{T}M.
When M is nonsingular, the Q and S factors of the polar decomposition are uniquely determined. However, the determinant of S is positive because S is positive definite, so Q inherits the sign of the determinant of M. That is, Q is only guaranteed to be orthogonal, not a rotation matrix. This is unavoidable; an M with negative determinant has no uniquely defined closest rotation matrix.
To efficiently construct a rotation matrix Q from an angle θ and a unit axis u, we can take advantage of symmetry and skewsymmetry within the entries. If x, y, and z are the components of the unit vector representing the axis, and
then
Determining an axis and angle, like determining a quaternion, is only possible up to sign; that is, (u,θ) and (−u,−θ) correspond to the same rotation matrix, just like q and −q. As well, axisangle extraction presents additional difficulties. The angle can be restricted to be from 0° to 180°, but angles are formally ambiguous by multiples of 360°. When the angle is zero, the axis is undefined. When the angle is 180°, the matrix becomes symmetric, which has implications in extracting the axis. Near multiples of 180°, care is needed to avoid numerical problems: in extracting the angle, a twoargument arctangent with atan2(sin θ,cos θ) equal to θ avoids the insensitivity of arccosine; and in computing the axis magnitude in order to force unit magnitude, a bruteforce approach can lose accuracy through underflow (Moler & Morrison 1983).
A partial approach is as follows:
The x, y, and z components of the axis would then be divided by r. A fully robust approach will use different code when t, the trace of the matrix Q, is negative, as with quaternion extraction. When r is zero because the angle is zero, an axis must be provided from some source other than the matrix.
Complexity of conversion escalates with Euler angles (used here in the broad sense). The first difficulty is to establish which of the twentyfour variations of Cartesian axis order we will use. Suppose the three angles are θ_{1}, θ_{2}, θ_{3}; physics and chemistry may interpret these as
while aircraft dynamics may use
One systematic approach begins with choosing the rightmost axis. Among all permutations of (x,y,z), only two place that axis first; one is an even permutation and the other odd. Choosing parity thus establishes the middle axis. That leaves two choices for the leftmost axis, either duplicating the first or not. These three choices gives us 3×2×2 = 12 variations; we double that to 24 by choosing static or rotating axes.
This is enough to construct a matrix from angles, but triples differing in many ways can give the same rotation matrix. For example, suppose we use the zyz convention above; then we have the following equivalent pairs:
(90°,  45°,  −105°)  ≡  (−270°,  −315°,  255°)  multiples of 360° 
(72°,  0°,  0°)  ≡  (40°,  0°,  32°)  singular alignment 
(45°,  60°,  −30°)  ≡  (−135°,  −60°,  150°)  bistable flip 
Angles for any order can be found using a concise common routine (Herter & Lott 1993; Shoemake 1994).
The problem of singular alignment, the mathematical analog of physical gimbal lock, occurs when the middle rotation aligns the axes of the first and last rotations. It afflicts every axis order at either even or odd multiples of 90°. These singularities are not characteristic of the rotation matrix as such, and only occur with the usage of Euler angles.
The singularities are avoided when considering and manipulating the rotation matrix as orthonormal row vectors (in 3D applications often named 'right'vector, 'up'vector and 'out'vector) instead of as angles. The singularities are also avoided when working with quaternions.
We sometimes need to generate a uniformly distributed random rotation matrix. It seems intuitively clear in two dimensions that this means the rotation angle is uniformly distributed between 0 and 2π. That intuition is correct, but does not carry over to higher dimensions. For example, if we decompose 3×3 rotation matrices in axisangle form, the angle should not be uniformly distributed; the probability that (the magnitude of) the angle is at most θ should be ^{1}⁄_{π}(θ − sin θ), for 0 ≤ θ ≤ π.
Since SO(n) is a connected and locally compact Lie group, we have a simple standard criterion for uniformity, namely that the distribution be unchanged when composed with any arbitrary rotation (a Lie group "translation"). This definition corresponds to what is called Haar measure. León, Massé & Rivest (2006) show how to use the Cayley transform to generate and test matrices according to this criterion.
We can also generate a uniform distribution in any dimension using the subgroup algorithm of Diaconis & Shashahani (1987). This recursively exploits the nested dimensions group structure of SO(n), as follows. Generate a uniform angle and construct a 2×2 rotation matrix. To step from n to n+1, generate a vector v uniformly distributed on the nsphere, S^{n}, embed the n×n matrix in the next larger size with last column (0,…,0,1), and rotate the larger matrix so the last column becomes v.
As usual, we have special alternatives for the 3×3 case. Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. Arvo (1992) takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation.
Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere. Since the homomorphism is a local isometry, we immediately conclude that to produce a uniform distribution on SO(3) we may use a uniform distribution on S^{3}.
Euler angles can also be used, though not with each angle uniformly distributed (Murnaghan 1962; Miles 1965).
For the axisangle form, the axis is uniformly distributed over the unit sphere of directions, S^{2}, while the angle has the nonuniform distribution over [0,π] noted previously (Miles 1965).