Appendix A : Linear Algebra

A.1 Definitions:

Matrix: m Rows and n Columns
Square Matrix(方阵): m == n
Column Vector(列向量): n == 1
Vector Component(向量分量): ai
Upper Triangular Matrix(上三角矩阵): a square matrix && if aij == 0 for i>j
Lower Triangular Matrix(下三角矩阵): a square matrix && if aij == 0 for i<j

Diagonal Matrix(对角矩阵): if aij  == 0 for i != j  

Identity Matrix In(单位矩阵): a square matrix && aii== 1
Null Matrix O(零矩阵): all elements are null
Null Column Vector 0
Transpose AT (转置): interchange rows and columns of the original matrix

                                   The Transpose of a column vector a is the row vector aT

Symmetric Matrix(对称矩阵): square matrix && AT  == A, thus aij ==  aji
Skew-Symmetric Matrix(反对称矩阵): square matrix && AT == − A, thus aij == − aji && aii == 0
Partitioned Matrix(分块矩阵): its elements are matrices of proper dimensions

Block-triangular Partitioned Matrix: similar to triangular matrix

Block-diagonal Partitioned Matrix: similar to diagonal matrix
Algebraic Element A(ij)(代数余子式): matrix of dimensions (n-1)*(n-1), obtained by 
                                   eliminating row i and column j of matrix A

A.2 Matrix Operations:

Trace Tr(A): the sum of the elements on the diagonal of a square matrix

A Equals to B: A and B are of same dimensions && aij = bij

The Sum of A and B: A and B are of same dimensions && C = A + B, thus cij =  aij +  bij
                                   Properties:

                                   A + O = A

                                     A + B = B + A

                                     (A + B) + C = A + (B + C)
                                   For partitioned matrix: if two matrices are of the same
                                   dimensions and partitioned in the same way, they 
                                   can be operated like elements.
Product of a scalar(标量) α by a matrix A αA: αaij 
                                   If A is a diagonal matrix && aii =  a: A = a In

If A is a square matrix: A =  As +  Aa

                                   Where As is symmetric part of A: As = 1/2( A +  AT )

                                      Aa is skew-symmetric part of A: Aa = 1/2( A −  AT )