Matrix algebra and transformations

An arithmetic for grids

A matrix is a rectangular array of numbers, added entry by entry when the shapes match. Multiplication is defined differently: the entry in row i, column j of AB comes from row i of A paired with column j of B, products summed. AB exists only when A has as many columns as B has rows.

Shapes are quoted rows first, so an m × n matrix has m rows and n columns, and is square when m and n agree and rectangular otherwise. Two matrices are equal when they have the same shape and match entry by entry. A pair of matrices whose shapes admit the product is called conformable, and the word carries the order with it, since AB can exist while BA does not. Nothing in any of this requires the entries to be real numbers. A matrix of complex entries is added, multiplied and inverted by the same rules, and one turns up as soon as the eigenvectors of a rotation are wanted.

The identity matrix I, with ones on the leading diagonal and zeros elsewhere, satisfies AI = IA = A. It is the identity element for matrix multiplication and is used to define the inverse.

The zero matrix 0, also called the null matrix, has every entry zero, so A + 0 = A and A0 = 0A = 0. One rule of ordinary arithmetic does not carry over: for numbers ab = 0 forces a = 0 or b = 0, but for matrices AB = 0 does not.

Matrix multiplication is associative: A(BC) = (AB)C whenever the shapes allow the products. That is why a chain of factors needs no brackets and why Aⁿ is unambiguous. Associativity permits the brackets to be moved; it does not permit the factors to be reordered.

Matrices as linear transformations of the plane

A 2 × 2 matrix acts on column vectors, and that action is a linear transformation: the origin is fixed and straight lines map to straight lines. A matrix represents a linear transformation through the images of the basis vectors, which form its columns: the first column is the image of i and the second is the image of j.

The standard matrices follow from that fact. Rotation anticlockwise by θ about the origin has columns (cos θ, sin θ) and (−sin θ, cos θ). Reflection in y = x swaps the unit vectors, so its columns are (0, 1) and (1, 0). Reflection in y = −x swaps them and negates both, giving columns (0, −1) and (−1, 0). Enlargement by scale factor k about the origin is k times the identity, and k is allowed to be negative.

A stretch parallel to the x-axis with scale factor a multiplies every x coordinate by a and leaves y unchanged. So i maps to (a, 0), j is fixed, and the matrix has rows (a, 0) and (0, 1). A stretch parallel to the y-axis has rows (1, 0) and (0, b). Applying both gives rows (a, 0) and (0, b), which is an enlargement only when a = b.

A shear fixes every point of one axis and translates every other point parallel to that axis by an amount proportional to its distance from it. With the x-axis invariant, i is fixed and j maps to (k, 1), so the matrix has rows (1, k) and (0, 1) and the transformation is (x, y) → (x + ky, y). A point on the x-axis is fixed, and a point one unit above it moves k to the right.

Describing a shear therefore requires two pieces of information: the invariant axis and the image of one point off it, which determines k. A shear with the x-axis invariant carrying (0, 1) to (3, 1) has k = 3. For a shear parallel to the y-axis the matrix has rows (1, 0) and (k, 1): the y-axis is invariant and (1, 0) maps to (1, k).

The determinant ad − bc gives the area scale factor as its absolute value, and its sign records whether orientation is preserved or reversed. The x-stretch has determinant a, so areas scale by |a|. Enlargement by k has determinant k², since both directions are scaled. Rotations have determinant 1 and reflections determinant −1: both preserve area, and the negative sign records that a reflection reverses orientation.

A shear has determinant 1 for every k, because the sheared unit square is a parallelogram on the same base with the same height. Determinant 1 therefore does not imply that lengths and angles are preserved: a rotation preserves both, while a shear with the same determinant changes both.

Composition is multiplication

Applying transformation B and then transformation A is the single transformation with matrix AB. The right-hand factor acts first, because it is adjacent to the vector in ABv. Since multiplication is generally non-commutative, a rotation followed by a reflection is generally a different transformation from the reflection followed by the rotation. Write the first-applied matrix on the right, not in the order the transformations are described.

Associativity makes a longer chain unambiguous. Applying C, then B, then A is the product ABC, and BC or AB may be evaluated first: the resulting transformation is the same. The grouping may be chosen for convenience; the order of the factors may not be changed.

Moving a curve rather than a point

A transformation acts on every point of the plane at once, so it acts on a curve as well. Reading off the image curve's equation means running backwards through the matrix. If (x, y) lies on the curve and M carries it to (X, Y), then (x, y) = M⁻¹(X, Y), and substituting those two expressions into the original equation leaves a relation between X and Y that the image points satisfy. It is the inverse that is substituted, for the same reason that replacing x by x − 3 in y = f(x) translates a graph three to the right.

The inverse-substitution method requires M to be invertible. If M is singular, parameterise the original curve and map its points directly; the image may be a line, a point or another lower-dimensional set.

The same substitution handles reflection in y = −x, whose matrix has rows (0, −1) and (−1, 0) and is its own inverse, so the image of a curve is found by writing −y for x and −x for y throughout. Under it y = x² becomes −x = y², a congruent parabola opening along the negative x-axis.

Transformations in three dimensions

Describe the drawing

Two axonometric sketches of the unit cube with the vectors i, j and k drawn from one corner and a three-by-three matrix above each: reflection in the plane x equals zero sends i to minus i and leaves j and k alone, while a quarter turn about the z-axis carries i onto the old j and j to minus i, and in both panels the columns of the matrix are exactly the marked images.

The 3 × 3 transformations required here form a short list, and each matrix is again read from the images of i, j and k. Reflection in the plane x = 0 negates the first coordinate, giving the diagonal matrix diag(−1, 1, 1), and reflections in y = 0 and z = 0 negate the second and third instead. Rotation about the z-axis leaves k alone and rotates the x and y coordinates, so it is the familiar 2 × 2 rotation sitting in the top-left corner with a 1 in the bottom-right.

Rotation about the x-axis is the same construction in the lower-right corner. For rotation about the y-axis the sine entries carry the opposite signs, with rows (cos θ, 0, sin θ), (0, 1, 0) and (−sin θ, 0, cos θ). State the sense convention being used for a rotation about an axis, such as the right-hand rule or the direction from which the rotation is viewed; the matrices above use the right-hand rule. Check a matrix by rotating a unit vector through 90° and comparing the image with that convention.