This class is a simple matrix class, it allows basic matrix operations, which are detailed below. The code here will be useful for linear algebra techniques which may arrive in graphics based programming. The matrix class gives the user the ability to test matrices for equality, add, subtract, multiply (by a scalar or another matrix), find an inverse, and easily access elements.
The matmain.cpp file contains unit tests used (UnitTest++) to test the code as it is being developed.
The + operator is overloaded to add two matrices of the same size. If an attempt to add two matrices of different sizes, an InvalidSize exception will be thrown.
The - operator has been overloaded to subtract two matrices, A and B (A-B). As with addition, if two matrices of differing sizes are subtracted, an exception will be thrown.
We have overridden () to index into our matrix in row major order (that is (a,b) accesses the element in row a, column b). The underlying data is not a two dimensional array and so we expect the user to use the () operator to get and change elements. If A is an mxn matrix, and we have i < 0 or i > m and j < 0 or j > n, then (i,j) will throw an exception.
We have overloaded * in two ways. For x, a real number, and A a matrix, we have xA is the matrix B such that B_{i,j} = xA_{i,j}. When multiplying two matrices A of size mxn and B of size kxl, we must have n=k, otherwise an exception will be thrown. The product, computed in the naive way produces an mxl matrix in time O(n^3).
We provide two methods for this, one multiplies the matrix by a 3 dimensional vector representing a direction vector, and the other represents multiplication by a 3 dimensional vector representing a point in 3-space. These both only work for a specif matrix size (4x4).
Determines whether two matrices A,B are equal. If the sizes aren't the same, it is clear that A!=B. Otherwise, we determine equality by checking each element in the matrices. We consider two elements equal if they are within 0.01 of eachother. That is, |A_{i,j}-B_{i,j}|<0.01, if this is not true for any (i,j) in {0,1,2,...,m-1}x{0,1,2,...,n-1}.
This method returns an nxn identity matrix.
Computes the inverse of some matrix A. If the matrix is not square, or is singular, an error occurs.
Creates the transpose of a matrix A_{i,j}. The transpose of a matrix is the matrix B=A_{j,i}.
Given a matrix we decompose into a product of simpler matrices. In LU, we simply get A=LU, where L is a lower diagonal matrix and U is an upper diagonal matrix. The LU decomposition does not always exist (even if A is nonsingular). For LUP decomposition, we find additionally a permutation matrix P such that PA=LU, this decomposition exists for all square nonsingular matrices.
Find the determinant of a square matrix A. We rely on an LUP decomposition to calculate this. Once the decomposition is found, we have L is unitary, and since we have L,U,P such that PA=LU, we see that Det(P)Det(A) = Det(PA) = Det(LU) = Det(L)Det(U) = Det(U). Since Det(P) = +/-1, we have Det(A) = +/-Det(U).
We use this algorithm to solve a lower triangular matrix
We use this algorithm to solve an upper triangular matrix
Given a matrix equation Ax=b, we wish to calculate x (we only handle the case where a unique solution exists). We decompose A into matrices L,U,P. Multiplying our original equation we see that LUx = PAx = Pb, and we now set y=Ux, and first solve Ly=Pb, then solve y=Ux using forward and barckwards substitution respectively.