
A matrix is composed of several rows and columns of numbers:
As shown in [1], a m x n (dimension) matrix has m rows and n columns. The elements of the matrix are denoted by a_{ij} where i = the row number, and j = the column number. In biomechanics and motion analysis, 3 x 3 and 3 x 1 matrices are the most commonly used types. A matrix of m = n is called a square matrix. In a diagonal matrix which is a square matrix, all elements except the diagonal ones (i = j) are zero:
Matrices of the same dimension can be added or subtracted, element by element:
As shown in [2], matrix addition is commutative. When a matrix is multiplied by a scalar, all the elements of the matrix magnify by the scalar value:
This operation is commutative as shown in [4]. Transpose of a m x n matrix is a n x m matrix whose columns are identical to the corresponding rows of the original matrix:
Note in [5] that the superscripted t , ()^{t}, is used as the symbol for the transpose. Transposing suffices the following:
One of the most useful properties of the matrix is the matrix multiplication. Imagine a system of linear equations:
where a's & c's = scalars, and x, y & z = the unknowns. [7] can be simplified as:
[8] is expressed in matrix multiplication form. The general form of matrix multiplication is
or
Note that the number of columns in the left vector (n in [9]) must be the same to the number of rows in the right matrix. The dimension of the resulting vector is m (rows of the left matrix) x p (columns of the right matrix) as shown in [9]. The matrix multiplication is not commutative:
but is distributive:
Similarly:
Matrix multiplication is also associative:
For any scalar d:
An identity matrix (I) is a square matrix whose diagonal elements are all 1 while the offdiagonal elements are all 0:
Then:
where d shown in [17] is the Kronecker delta:
From [5] and [9]:
The determinant of a 2 x 2 matrix is defined as:
The determinant of a 3 x 3 matrix can be reduced to a series of the determinants of 2 x 2 matrices:
To generalize [21] for any square matrix, a new matrix needs to be defined:
In other words, matrix a_{ij} is matrix a less the ith row and the jth column. Then, [21] can be generalize to
[23] is called a cofactor expansion across the first row of a. In fact, the determinant of a can be cofactorexpanded across any row or column:
The following properties of the determinant hold:
The inverse matrix of a square matrix suffices the following relationship:
where a^{1} = the inverse matrix of a, and I = the identity matrix. For a square matrix to be invertible, its determinant must not be 0. The inverse matrix of square matrix a can be expressed as
The matrix of cofactors of a on the right side in [27] is called the adjugate of a. Note in [27] that element a_{ij} of the adjugate is associated with det(a_{ji}) rather than det(a_{ij}). A system of linear equations such as that in [8] can be generalized as
where a = the known square coefficient matrix, b = the known column matrix, and x = the unknown parameter matrix. If matrix a is invertible, the unknown parameters can be obtained as follows:
In motion analysis, matrix a shown in [28] is generally not square. If matrix a is not a square matrix, the system of linear equations can be solved as
[30] is the socalled least square method. See Least Square Method for more details. 
© YoungHoo Kwon, 1998 