3D mathematics-more knowledge about matrices (2)

Matrix Inverse

Another important matrix operation is matrix inversion. This operation can only be used for square arrays.

 

Algorithm

PhalanxMAsM-1Is also a matrix. WhenMAndM-1The result is a matrix of units. Expressed as formula 9.6:

Not all matrices have inverse values. An obvious example is that if the element in a row or column of a matrix is 0 and any matrix is used to multiply the matrix, the result is a zero matrix. If a matrix has an inverse matrix, it is called reversible or non-singular. If a matrix does not have an inverse matrix, it is called an irreversible or singular matrix. The determining factor of a singular matrix is 0, and that of a non-singular matrix is not 0. Therefore, the value of the determining factor is an effective method for determining whether the matrix is reversible. In addition, for any reversible MatrixM, When and only whenV=0,VM=0.

M"Standard adjoint matrix" as "adjM", Is defined as the transpose matrix of the algebraic remainder matrix of M. The following is an example. Consider the 3x3 matrix given above.M:

ComputingMAlgebra of the remainder matrix:

MThe standard adjoint matrix of is the transpose of the algebraic remainder matrix:

Once the standard adjoint matrix is availableMTo calculate the inverse of the matrix.

The expression is shown in formula 9.7:

For example, to obtain the inverse of the above matrix, there are:

Of course, there are other methods that can be used to calculate the inverse of a matrix, such as Gaussian elimination method. Many linear algebra books have concluded that this method is more suitable for Computer Implementation, because it uses less algebraic operations, which is actually incorrect. This may be true for large matrices or some special matrices. However, for low-level matrices, such as those common low-level matrices in Geometric Applications, standard adjoint matrices may be faster. Because it can provide branchless implementation for the standard adjoint matrix, this implementation method is faster in today's over-the-Standard Architecture and dedicated Vector processors.

Important Properties of Matrix Inversion:

 

GEOMETRIC INTERPRETATION

The inverse of a matrix is very useful in ry, because it allows us to calculate the inverse or inverse transformation of a transformation-Can "undo" the original transformation. Therefore, if the VectorVMatrixMAnd then useMInverseM-1To obtain the original vector. This is easily verified by the algebraic method: