多元统计分析理论基础,矩阵和多元正态分布,双语

1、 Data Matrices and Manipulation zfPresentation Outline1、Matrices and vectors2、Random vector3、Multivariate data matrix4、Population mean vector, covariance and correlation matrices5、Sample mean vector, covariance and correlation matrices6、Euclidean, statistical and Mahalanobis distances zf21、 Matrices

2、 and vectorsA 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）zf3zf4zf5If n = p , then this is a square matrix（方阵）.If a matrix has only one column, then this is called a column vector（列向量）.If there is only one row, then this is called a row vecto

3、r.（行向量）zf6Transpose of a Matrix（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:zf7例：给定一个矩阵A，矩阵A的转置矩阵是？zf8其他特殊矩阵形式和定义：A zero matrix has all elements equal to zero.（零矩阵） A diagonal matrix is a square matrix that has element

4、s of zero, except down the main diagonal.（对角矩阵）zf9A symmetric matrix is a square matrix that is unchanged when it is transposed I.e. A=A.（对称矩阵矩阵的转置和它本身相等）zf10An identity matrix (I) is a diagonal matrix with all diagonal terms being unity.（单位矩阵）zf11An inverse matrix （逆矩阵） To a square matrix A, if a s

5、quare matrix B exists and AB=BA=I. Then B is the inverse matrix of A(or A is the inverse matrix of B)zf12The trace of a matrix is the sum of the diagonal terms （矩阵的迹）. This is only defined for square matrices.例：给定一个矩阵A，求矩阵A的迹？ tr(A)= =a+bzf13矩阵运算 （1）Matrix Addition and Subtraction（矩阵加法和减法） 例： zf14续例

6、1： 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。Matrices with different dimensions cannot be added or subtracted.（只有当两个矩阵同行数、同列数时，才能相加减。）zf15（2）Matrix Multiplication（矩阵乘法）Scalar Multiplication（数乘运算）： 续例1：求每人每科两次考试的平均成绩zf16Matrix Multiplication（矩阵乘法）: To multiply two matrices, the column dimension of the matrix on

7、the left must equal the row dimension of the matrix on the right（两个矩阵相乘，第一个矩阵的列数必须等于第二个矩阵的行数）.zf17zf18zf19（3）Matrix Multiplication Algebra（矩阵乘法的代数式） (AB) = 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. zf20Determi

8、nant and Inverse of a Matrix（矩阵行列式和逆矩阵）：1、Determinant of a matrix（矩阵行列式）:zf212、The inverse of a matrix（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 A is the inverse matrix of B)zf22For a square matrix A, we may find its inverse such that =

9、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0，则该方阵无逆矩阵）. A matrix with a zero determinant is described as being singular（如果方阵的行列式等于0，则称该方阵为奇异矩阵；否则，为非奇异矩阵nonsingular ）.A matrix A is orthogonal if and only if （如果A的逆矩阵等于其转置矩阵，则称矩阵A正交）zf23二阶逆矩阵运算：zf24例：zf25Eig

10、envalues and Eigenvectors（向量与特征向量）：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）The equation is called the characteristic equation(特征方程）.If C is a nonzero vector such that AC= 或 then C is said to be an eigenvector (特征向量）of the matrix A associated with the eigenvalue .The sum of the eigenvalues of A is equal to the trace of Azf26zf27zf28zf29例：Find the eigenvalues and eigenvectors ofzf30zf312、Random Vector（随机向量）zf323、Multivariate Data Matrix(多元数据矩阵）zf334、mean vector, co