多元统计分析理论基础,矩阵和多元正态分布,双语

1、zfData Matrices and Manipulation2021-11-192cxtPresentation Outlinev 1、Matrices and vectorsv 2、Random vectorv 3、Multivariate data matrixv 4、Population mean vector, covariance and correlation matricesv 5、Sample mean vector, covariance and correlation matricesv 6、Euclidean, statistical and Mahalanobis

2、distances 2021-11-193cxt1、 Matrices and vectorsv A matrix of size n p is a rectangular array of numbers with n rows and p columns of the form（ n 行 p列 矩阵矩阵）2021-11-194cxt2021-11-195cxt2021-11-196cxtv If n = p , then this is a square matrix（方阵方阵）.v If a matrix has only one column, then this is called

3、a column vector（列向量列向量）.v If there is only one row, then this is called a row vector.（行向量行向量）2021-11-197cxtv Transpose of a Matrix（转置矩阵）转置矩阵） denoted by a prime,is found by interchanging the rows and the columns.（将矩阵的行和列交换） The transpose of A, C and R above are:2021-11-198cxtv 例：例：给定一个矩阵A A，矩阵A A的的转

4、置矩阵是？2021-11-199cxt其他特殊矩阵形式和定义：其他特殊矩阵形式和定义：v A zero matrix has all elements equal to zero.（零零矩阵矩阵） v A diagonal matrix is a square matrix that has elements of zero, except down the main diagonal.（对角矩阵对角矩阵）0000000001000200052021-11-1910cxtv A symmetric matrix is a square matrix that is unchanged when

5、 it is transposed I.e. A=A.（对称矩对称矩阵阵矩阵的转置和它本身相等）2021-11-1911cxtv An identity matrix (I) is a diagonal matrix with all diagonal terms being unity.（单位矩阵）单位矩阵）2021-11-1912cxtv An inverse matrix （逆矩阵）逆矩阵） To a square matrix A, if a square matrix B exists and AB=BA=I. Then B is the inverse matrix of A(or

6、 A is the inverse matrix of B)2021-11-1913cxtv The trace of a matrix is the sum of the diagonal terms （矩阵的迹）矩阵的迹）. This is only defined for square matrices.v 例：给定一个矩阵A A，v 求矩阵A的迹？ tr(A)= =a+b1iipia2021-11-1914cxtv 矩阵运算矩阵运算 （1 1）Matrix Addition and Subtraction（矩矩阵加法和减法）阵加法和减法） 例： 2021-11-1915cxtv 续例续

7、例1 1： v 欲求每人、每科两次考试的总分数，即把两个矩阵的对应元素相加。v Matrices with different dimensions cannot be added or subtracted.（只有当两个矩阵同行数、只有当两个矩阵同行数、同列数时，才能相加减。）同列数时，才能相加减。）2021-11-1916cxtv （2）Matrix Multiplication（矩阵乘法）矩阵乘法）v Scalar Multiplication（数乘运算）数乘运算）： 续例1：求每人每科两次考试的平均成绩2021-11-1917cxtv Matrix Multiplication（矩阵乘

8、法矩阵乘法）: v To multiply two matrices, the column dimension of the matrix on the left must equal the row dimension of the matrix on the right（两个矩阵相乘，第一个矩阵的列数两个矩阵相乘，第一个矩阵的列数必须等于第二个矩阵的行数必须等于第二个矩阵的行数）.2021-11-1918cxt2021-11-1919cxt2021-11-1920cxtv （3）Matrix Multiplication Algebra（矩阵乘法的矩阵乘法的代数式）代数式） (AB) =

9、 BA (Note reversal of positions) In general, AB is not equal to BA. AB = 0 does not imply A= 0 or B = 0 If A = 0 or B =0 then AB = 0. 2021-11-1921cxtDeterminant and Inverse of a Matrix（矩阵行列式和逆矩阵）：矩阵行列式和逆矩阵）：v 1、Determinant of a matrix（矩阵行列式矩阵行列式）:2021-11-1922cxtv 2、The inverse of a matrix（逆矩阵）逆矩阵） T

10、o a square matrix A, if a square matrix B exists and AB=BA=I. Then B is the inverse matrix of A(or A is the inverse matrix of B)2021-11-1923cxtv For a square matrix A, we may find its inverse such that = I. The inverse of a matrix is not defined if its determinant is equal to zero（如果方阵的行列式等于如果方阵的行列式

11、等于0 0，则该方，则该方阵无逆矩阵阵无逆矩阵）. A matrix with a zero determinant is described as being singular（如果方阵的行列式等于如果方阵的行列式等于0 0，则称该方阵为，则称该方阵为奇异矩阵；否则，为非奇异矩阵奇异矩阵；否则，为非奇异矩阵nonnonsingular ）.A matrix A is orthogonal if and only if （如果如果A A的的逆矩阵等于其转置矩阵，则称矩阵逆矩阵等于其转置矩阵，则称矩阵A A正交正交）2021-11-1924cxtv 二阶逆矩阵运算：2021-11-1925cxt

12、例：例：2021-11-1926cxtEigenvalues and Eigenvectors（向量与特征向量）：向量与特征向量）：v Let A be an n n square matrix and I be the n n identity matrix. Then the scalars satisfying the polynomial equation are called eigenvalues of a matrix A（特征值）特征值）v The equation is called the characteristic equation( (特征方程）特征方程）.v If

13、C is a nonzero vector such that AC= 或 then C is said to be an eigenvector (特征向量特征向量）of the matrix A associated with the eigenvalue .v The sum of the eigenvalues of A is equal to the trace of AC()0AI C2021-11-1927cxt2021-11-1928cxt2021-11-1929cxt2021-11-1930cxtv 例：Find the eigenvalues and eigenvectors of2021-11-1931cxt2021-11-1932cxt2、Random Vector（随机向量）随机向量）2021-11-1933cxt3、Multivariate Data Matrix( (多元数据矩阵）多元数据矩阵）2021-11-1934cxt4、mean vector, covariance and correlation matrices（均值向量、协方差、相关系数矩阵）总体均值向量、协方差、相关系数矩阵）总体2021-11-1935cxt2021-11-1936