22 APPENDIX 1: MATH FACTS 22.1 Vectors 22.1.1 Definition A vector has a dual definition: It is a segment of a a line with direction, or it consists of its projection on a reference system 0xyz, usually orthogonal and right handed. The first form is independent of any reference system, whereas the second (in terms of its components) depends directly on the coordinate system. Here we use the second notation, i.e., x is meant as a column vector, whose components are found as projections of an (invariant) directed segment on a specific reference system. We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms of its components with respect to a reference system as ⎡ ⎧⎢ ⎤ ⎧⎥ 2 1 γa = . ⎧⎣⎧⎨7 The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. z 2 x y 1 7 a22.1 Vectors 111 1. Vector addition: γa + γb = γc ⎡ ⎧⎢ 2 ⎡ ⎧⎢ ⎤ ⎧⎥ 3 ⎡ ⎧⎢ ⎤ ⎧⎥ 5 ⎤ ⎧⎥ 1+3 = 4 . ⎧⎣7 ⎧⎣⎧⎨2 ⎧⎣⎧⎨9 ⎧⎨ Graphically, addition is stringing the vectors together head to tail. 2. Scalar multiplication: ⎡ ⎧⎢ ⎡ ⎧⎢ ⎤ ⎧⎥ ⎤ ⎧⎥ 2 −4 −2 × 1 = −2 . ⎧⎣⎧⎣⎧⎨⎧⎨7 −14 22.1.2 Vector Magnitude The total length of a vector of dimension m, its Euclidean norm, is given by m ⎫ ⎦���2||γx|| = xi . i=1 This scalar is commonly used to normalize a vector to length one. 22.1.3 Vector Dot or Inner Product The dot product of two vectors is a scalar equal to the sum of the products of the corre-sponding components: m ⎫ x γ y =γ y = γx T γ· xiyi. i=1 The dot product also satisfies γx γ· y = y|| cos χ, ||γx||||γwhere χ is the angle between the vectors.� � � � � � � � � � � � � � � 112 22 APPENDIX 1: MATH FACTS 22.1.4 Vector Cross Product x and γy is another vector γz, γ yThe cross product of two three-dimensional vectors γ x × γ = γz, whose 1. direction is normal to the plane formed by the other two vectors, 2. direction is given by the right-hand rule, rotating from γx to γy, 3. magnitude is the area of the parallelogram formed by the two vectors – the cross product of two parallel vectors is zero – and 4. (signed) magnitude is equal to ||γ y|| sin χ, where χ is the angle between the two x||||γvectors, measured from γx to γy. In terms of their components, ⎡ ⎧⎧⎢ ⎤ ⎧⎧⎥ γx × γy = ˆˆˆi j k x1 x2 x3 y1 y2 y3 = (x2y3 − x3y2)ˆi j (x1y2 − x2y1)ˆ(x3y1 − x1y3)ˆk . ⎧⎧⎣ ⎧⎧⎨ 22.2 Matrices 22.2.1 Definition A matrix, or array, is equivalent to a set of column vectors of the same dimension, arranged side by side, say ⎭2 3 A = [γa γb ] = ⎛⎝1 3 ⎞⎠. 7 2 This matrix has three rows (m = 3) and two columns (n = 2); a vector is a special case of a matrix with one column. Matrices, like vectors, permit addition and scalar multiplication. We usually use an upper-case symbol to denote a matrix. 22.2.2 Multiplying a Vector by a Matrix If Aij denotes the element of matrix A in the i’th row and the j’th column, then the multi-plication γc = Aγv is constructed as: n ⎫ ci = Ai1v1 + Ai2v2 + + Ainvn· · · = Aij vj , j=1 where n is the number of columns in A. γc will have as many rows as A has rows (m). Note that this multiplication is defined only if γv has as many rows as A has columns; they have� � 22.2 Matrices 113 consistent inner dimension n. The product γvA would be well-posed only if A had one row, and the proper number of columns. There is another important interpretation of this vector multiplication: Let the subscript : indicate all rows, so that each A:j is the j’th column vector. Then γc = Aγv = A:1v1 + A:2v2 + + A:nvn.···We are multiplying column vectors of A by the scalar elements of γv. 22.2.3 Multiplying a Matrix by a Matrix The multiplication C = AB is equivalent to a side-by-side arrangement of column vectors C:j = AB:j , so that C = AB = [AB:1 AB:2 AB:k],··· where k is the number of columns in matrix B. The same inner dimension condition applies as noted above: the number of columns in A must equal the number of rows in B. Matrix multiplication is: 1. Associative. (AB)C = A(BC). 2. Distributive. A(B + C) = AB + AC, (B + C)A = BA + CA. 3. NOT Commutative. AB = BA, except in special cases. ≥22.2.4 Common Matrices Identity. The identity matrix is usually denoted I, and comprises a square matrix with ones on the diagonal, and zeros elsewhere, e.g., ⎭1 0 0 0 1 0 ⎛⎝ ⎞⎠I3= ×3 . 0 0 1 The identity always satisfies AIn×n A = A.= Im×mDiagonal Matrices. A diagonal matrix is square, and has all zeros off the diagonal. For instance, the following is a diagonal matrix: ⎭4 0 0 ⎛⎝ ⎞⎠A = 0 −2 0 0 . 0 3 The product of a diagonal matrix with another diagonal matrix is diagonal, and in this case the operation is commutative.� � 114 22 APPENDIX 1: MATH FACTS 22.2.5 Transpose The transpose of a vector or matrix, indicated by a T superscript results from simply swap-ping the row-column indices of each entry; it is equivalent to “flipping” the vector or matrix around the diagonal line. For example, ⎡ ⎧⎢ 1 ⎤ ⎧⎥ a−∀ γT = { 1 2 3}γa = 2 ⎧⎣3 ⎧⎨ ⎬ ⎭1 2 1 4 8 ⎞⎠−∀ ATA = ⎛⎝4 5 = . 2 5 9 8 9 A very useful property of the transpose is (AB)T = BT AT . 22.2.6 Determinant The determinant of a square matrix A is a scalar equal to the volume of the parallelepiped enclosed by the constituent vectors. The two-dimensional case is particularly easy to re-member, and illustrates the principle of volume: det(A) = A11A22 − A21A12 �⎬ ��1 − 1 det = 1 + 1 = 2. 1 1 1-1 A:2 A:1 Area = det(A) = 2 y x In higher dimensions, the determinant is more complicated to compute. The general formula allows one to pick a row k, perhaps the one containing the most zeros, and apply det(A) = j=n ⎫ j=1 Akj(− 1)k+j�kj , where �kj is the determinant of the sub-matrix formed by neglecting the k’th row and the j’th column. The formula is symmetric, in the sense that one could also target the k’th column:� 22.2 Matrices 115 det(A) = j=n ⎫ j=1 Ajk(−1)k+j �jk . If the determinant of a matrix is zero, then the matrix is said