Data Matrices and Manipulation
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Presentation Outline
1、Matrices and vectors 2、Random vector 3、Multivariate data matrix 4、Population mean vector, covariance and correlation matrices 5、Sample mean vector, covariance and correlation matrices 6、Euclidean, statistical and Mahalanobis distances
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An identity matrix (I) is a diagonal matrix with all diagonal terms being unity.（单位矩阵）
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An inverse 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)
If there is only one row, then this is called a row vector.（行向量）
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Transpose 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:
1 0 0 0 2 0 0 0 5 0 0 0 0 0 0
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A 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）
p i 1
ii
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矩阵运算 （1）Matrix Addition and Subtraction（矩 阵加法和减法） 例：
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续例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。）
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Determinant and Inverse of a Matrix（矩阵行列式和逆矩阵）：
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（3）Matrix Multiplication Algebra（矩阵乘法的 代数式） (AB)’ = B’A’ (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.
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（2）Matrix Multiplication（矩阵乘法）
Scalar Multiplication（数/21
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Matrix Multiplication（矩阵乘法）:
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 必须等于第二个矩阵的行数）.
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1、 Matrices and vectors
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）
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The 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 =a+b
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If 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）.
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例：给定一个矩阵A，
矩阵A的转置矩阵是？？
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其他特殊矩阵形式和定义：
A zero matrix has all elements equal to zero.（零 0 0 0 矩阵） A diagonal matrix is a square matrix that has elements of zero, except down the main diagonal. （对角矩阵）