Inlinear algebra,linear transformations can be represented bymatrices. If${\displaystyle T}$ is a linear transformation mapping${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle {\mathbb{R}}^m}$ and${\displaystyle \mathbf{x}}$ is acolumn vector with${\displaystyle n}$ entries, then there exists an${\displaystyle m\times n}$ matrix${\displaystyle A}$, called thetransformation matrix of${\displaystyle T}$,^[1] such that:$${\displaystyle T(\mathbf{x})=A\mathbf{x}}$$Note that${\displaystyle A}$ has${\displaystyle m}$ rows and${\displaystyle n}$ columns, whereas the transformation${\displaystyle T}$ is from${\displaystyle {\mathbb{R}}^n}$ to${\displaystyle {\mathbb{R}}^m}$. There are alternative expressions of transformation matrices involvingrow vectors that are preferred by some authors.^[2][3]

Matrices allow arbitrarylinear transformations to be displayed in a consistent format, suitable for computation.^[1] This also allows transformations to becomposed easily (by multiplying their matrices).

Linear transformations are not the only ones that can be represented by matrices. Some transformations that are non-linear on an n-dimensionalEuclidean spaceRⁿ can be represented as linear transformations on then+1-dimensional spaceRⁿ⁺¹. These include bothaffine transformations (such astranslation) andprojective transformations. For this reason, 4×4 transformation matrices are widely used in3D computer graphics. Thesen+1-dimensional transformation matrices are called, depending on their application,affine transformation matrices,projective transformation matrices, or more generallynon-linear transformation matrices. With respect to ann-dimensional matrix, ann+1-dimensional matrix can be described as anaugmented matrix.

In thephysical sciences, anactive transformation is one which actually changes the physical position of asystem, and makes sense even in the absence of acoordinate system whereas apassive transformation is a change in the coordinate description of the physical system (change of basis). The distinction between active and passivetransformations is important. By default, bytransformation,mathematicians usually mean active transformations, whilephysicists could mean either.

Put differently, apassive transformation refers to description of thesame object as viewed from two different coordinate frames.

If one has a linear transformation${\displaystyle T(x)}$ in functional form, it is easy to determine the transformation matrixA by transforming each of the vectors of thestandard basis byT, then inserting the result into the columns of a matrix. In other words,

For example, the function${\displaystyle T(x)=5x}$ is a linear transformation. Applying the above process (suppose thatn = 2 in this case) reveals that:

The matrix representation of vectors and operators depends on the chosen basis; asimilar matrix will result from an alternate basis. Nevertheless, the method to find the components remains the same.

To elaborate, vector${\displaystyle \mathbf{v}}$can be represented in basis vectors, with coordinates:$${\displaystyle \mathbf{v}=v_1{\mathbf{e}}_1+v_2{\mathbf{e}}_2+\cdots +v_n{\mathbf{e}}_n=\sum _i{v}_i{\mathbf{e}}_i=E[\mathbf{v}{]}_E}$$

Now, express the result of the transformation matrixA upon${\displaystyle \mathbf{v}}$, in the given basis:

The${\displaystyle a_{i,j}}$ elements of matrixA are determined for a given basisE by applyingA to every, and observing the response vector$${\displaystyle A{\mathbf{e}}_j=a_{1,j}{\mathbf{e}}_1+a_{2,j}{\mathbf{e}}_2+\cdots +a_{n,j}{\mathbf{e}}_n=\sum _i{a}_{i,j}{\mathbf{e}}_i.}$$

This equation defines the wanted elements,${\displaystyle a_{i,j}}$, ofj-th column of the matrixA.^[4]

Yet, there is a special basis for an operator in which the components form adiagonal matrix and, thus, multiplication complexity reduces ton. Being diagonal means that all coefficients${\displaystyle a_{i,j}}$ except${\displaystyle a_{i,i}}$ are zeros leaving only one term in the sum$\sum a_{i,j}{\mathbf{e}}_i$ above. The surviving diagonal elements,${\displaystyle a_{i,i}}$, are known aseigenvalues and designated with${\displaystyle {\lambda }_i}$ in the defining equation, which reduces to${\displaystyle A{\mathbf{e}}_i={\lambda }_i{\mathbf{e}}_i}$. The resulting equation is known aseigenvalue equation.^[5] Theeigenvectors and eigenvalues are derived from it via thecharacteristic polynomial.

Withdiagonalization, it isoften possible totranslate to and from eigenbases.

Most commongeometric transformations that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear. In two dimensions, linear transformations can be represented using a 2×2 transformation matrix.

A stretch in thexy-plane is a linear transformation which enlarges all distances in a particular direction by a constant factor but does not affect distances in the perpendicular direction. We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the formx' =kx;y' =y for some positive constantk. (Note that ifk > 1, then this really is a "stretch"; ifk < 1, it is technically a "compression", but we still call it a stretch. Also, ifk = 1, then the transformation is an identity, i.e. it has no effect.)

The matrix associated with a stretch by a factork along the x-axis is given by:

Similarly, a stretch by a factork along the y-axis has the formx' =x;y' =ky, so the matrix associated with this transformation is

If the two stretches above are combined with reciprocal values, then the transformation matrix represents asqueeze mapping:A square with sides parallel to the axes is transformed to a rectangle that has the same area as the square. The reciprocal stretch and compression leave the area invariant.

Forrotation by an angle θcounterclockwise (positive direction) about the origin the functional form is${\displaystyle x^{\prime }=x\cos \theta -y\sin \theta }$ and${\displaystyle y^{\prime }=x\sin \theta +y\cos \theta }$. Written in matrix form, this becomes:^[6]

Similarly, for a rotationclockwise (negative direction) about the origin, the functional form is${\displaystyle x^{\prime }=x\cos \theta +y\sin \theta }$ and${\displaystyle y^{\prime }=-x\sin \theta +y\cos \theta }$ the matrix form is:

These formulae assume that thex axis points right and they axis points up.

Forshear mapping (visually similar to slanting), there are two possibilities.

A shear parallel to thex axis has${\displaystyle x^{\prime }=x+ky}$ and${\displaystyle y^{\prime }=y}$. Written in matrix form, this becomes:

A shear parallel to they axis has${\displaystyle x^{\prime }=x}$ and${\displaystyle y^{\prime }=y+kx}$, which has matrix form:

For reflection about a line that goes through the origin, let${\displaystyle \mathbf{l}=(l_x,l_y)}$ be avector in the direction of the line. Then the transformation matrix is:

To project a vector orthogonally onto a line that goes through the origin, let${\displaystyle \mathbf{u}=(u_x,u_y)}$ be avector in the direction of the line. Then the transformation matrix is:

As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation.

Parallel projections are also linear transformations and can be represented simply by a matrix. However, perspective projections are not, and to represent these with a matrix,homogeneous coordinates can be used.

Thematrix to rotate an angleθ about any axis defined byunit vector (x,y,z) is^[7]

To reflect a point through a plane${\displaystyle ax+by+cz=0}$ (which goes through the origin), one can use${\displaystyle \mathbf{A}=\mathbf{I}-2{\mathbf{N}\mathbf{N}}^{\mathrm{T}}}$, where${\displaystyle \mathbf{I}}$ is the 3×3 identity matrix and${\displaystyle \mathbf{N}}$ is the three-dimensionalunit vector for the vector normal of the plane. If theL² norm of${\displaystyle a}$,${\displaystyle b}$, and${\displaystyle c}$ is unity, the transformation matrix can be expressed as:

Note that these are particular cases of aHouseholder reflection in two and three dimensions. A reflection about a line or plane that does not go through the origin is not a linear transformation — it is anaffine transformation — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector):where${\displaystyle d=-\mathbf{p}\cdot \mathbf{N}}$ for some point${\displaystyle \mathbf{p}}$ on the plane, or equivalently,${\displaystyle ax+by+cz+d=0}$.

If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix. Seehomogeneous coordinates andaffine transformations below for further explanation.

One of the main motivations for using matrices to represent linear transformations is that transformations can then be easilycomposed and inverted.

Composition is accomplished bymatrix multiplication.Row and column vectors are operated upon by matrices, rows on the left and columns on the right. Since text reads from left to right, column vectors are preferred when transformation matrices are composed:

IfA andB are the matrices of two linear transformations, then the effect of first applyingA and thenB to a column vector${\displaystyle \mathbf{x}}$ is given by:$${\displaystyle \mathbf{B}(\mathbf{A}\mathbf{x})=(\mathbf{B}\mathbf{A})\mathbf{x}.}$$

In other words, the matrix of the combined transformationA followed byB is simply the product of the individual matrices.

WhenA is aninvertible matrix there is a matrixA⁻¹ that represents a transformation that "undoes"A since its composition withA is theidentity matrix. In some practical applications, inversion can be computed using general inversion algorithms or by performing inverse operations (that have obvious geometric interpretation, like rotating in opposite direction) and then composing them in reverse order. Reflection matrices are a special case becausethey are their own inverses and don't need to be separately calculated.

To representaffine transformations with matrices, we can usehomogeneous coordinates. This means representing a 2-vector (x,y) as a 3-vector (x,y, 1), and similarly for higher dimensions. Using this system, translation can be expressed with matrix multiplication. The functional form${\displaystyle x^{\prime }=x+t_x;y^{\prime }=y+t_y}$ becomes:

All ordinary linear transformations are included in the set of affine transformations, and can be described as a simplified form of affine transformations. Therefore, any linear transformation can also be represented by a general transformation matrix. The latter is obtained by expanding the corresponding linear transformation matrix by one row and column, filling the extra space with zeros except for the lower-right corner, which must be set to 1. For example,thecounter-clockwiserotation matrix from above becomes:

Using transformation matrices containing homogeneous coordinates, translations become linear, and thus can be seamlessly intermixed with all other types of transformations. The reason is that the real plane is mapped to thew = 1 plane in real projective space, and so translation in realEuclidean space can be represented as a shear in real projective space. Although a translation is a non-linear transformation in a 2-D or 3-D Euclidean space described by Cartesian coordinates (i.e. it can't be combined with other transformations while preservingcommutativity and other properties), itbecomes, in a 3-D or 4-D projective space described by homogeneous coordinates, a simple linear transformation (ashear).

More affine transformations can be obtained bycomposition of two or more affine transformations. For example, given a translationT' with vector${\displaystyle (t_x^{\prime },t_y^{\prime }),}$ a rotationR by an angle θcounter-clockwise, a scalingS with factors${\displaystyle (s_x,s_y)}$ and a translationT of vector${\displaystyle (t_x,t_y),}$ the resultM ofT'RST is:^[8]

When using affine transformations, the homogeneous component of a coordinate vector (normally calledw) will never be altered. One can therefore safely assume that it is always 1 and ignore it. However, this is not true when using perspective projections.

Another type of transformation, of importance in3D computer graphics, is theperspective projection. Whereas parallel projections are used to project points onto the image plane along parallel lines, the perspective projection projects points onto the image plane along lines that emanate from a single point, called the center of projection. This means that an object has a smaller projection when it is far away from the center of projection and a larger projection when it is closer (see alsoreciprocal function).

The simplest perspective projection uses the origin as the center of projection, and the plane at${\displaystyle z=1}$ as the image plane. The functional form of this transformation is then${\displaystyle x^{\prime }=x/z}$;${\displaystyle y^{\prime }=y/z}$. We can express this inhomogeneous coordinates as:

After carrying out thematrix multiplication, the homogeneous component${\displaystyle w_c}$ will be equal to the value of${\displaystyle z}$ and the other three will not change. Therefore, to map back into the real plane we must perform thehomogeneous divide orperspective divide by dividing each component by${\displaystyle w_c}$:$${\displaystyle \left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\\ 1\end{array}\right]=\frac{1}{w_c}\left[\begin{array}{c}{x}_c\\ y_c\\ z_c\\ w_c\end{array}\right]=\left[\begin{array}{c}x/z\\ y/z\\ 1\\ 1\end{array}\right]}$$

More complicated perspective projections can be composed by combining this one with rotations, scales, translations, and shears to move the image plane and center of projection wherever they are desired.

• 3D projection
• Change of basis
• Image rectification
• Pose (computer vision)
• Rigid transformation
• Transformation (function)
• Transformation geometry

• The Matrix Page Practical examples inPOV-Ray
• Reference page - Rotation of axes
• Linear Transformation Calculator
• Transformation Applet - Generate matrices from 2D transformations and vice versa.
• Coordinate transformation under rotation in 2D
• Excel Fun - Build 3D graphics from a spreadsheet

• Computer graphics
• Matrices
• Transformation (function)

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