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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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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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? 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 ）,
? 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,
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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
矩阵 ）
? A diagonal matrix is a square matrix that has
elements of zero,except down the main diagonal.
（ 对角矩阵 ）
000
000
000
??
??
????
1 0 0
0 2 0
0 0 5
??
??
????
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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）
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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)
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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+b
1 ii
p
i
a
?
?
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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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? （ 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
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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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? （ 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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Determinant and Inverse of a Matrix（ 矩阵行列式和逆矩阵）,
? 1,Determinant of a matrix（ 矩阵行列式 ）,
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? 2,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)
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? 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 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 ）
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? 二阶逆矩阵运算,
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例,
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Eigenvalues 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 A
? ?
C? ( ) 0A I C???
?
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? 例,Find the eigenvalues and eigenvectors of
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2,Random Vector（ 随机向量）
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3,Multivariate Data Matrix(多元数据矩阵）
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4,mean vector,covariance and correlation matrices
（ 均值向量、协方差、相关系数矩阵）－总体
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4,mean vector,covariance and correlation matrices
（ 均值向量、协方差、相关系数矩阵）－样本
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5、欧氏距离与马氏距离
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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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 ）
?
2011-10-16 4
cxt
2011-10-16 5
cxt
2011-10-16 6
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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 ）,
? If there is only one row,then this is called a
row vector.（ 行向量 ）
2011-10-16 7
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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,
2011-10-16 8
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? 例,给定一个矩阵 A,
矩阵 A的 转置矩阵是？？
2011-10-16 9
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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
矩阵 ）
? A diagonal matrix is a square matrix that has
elements of zero,except down the main diagonal.
（ 对角矩阵 ）
000
000
000
??
??
????
1 0 0
0 2 0
0 0 5
??
??
????
2011-10-16 10
cxt
? 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）
2011-10-16 11
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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)
2011-10-16 13
cxt
? 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+b
1 ii
p
i
a
?
?
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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。）
2011-10-16 16
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? （ 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
2011-10-16 17
cxt
? 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 ）,
2011-10-16 18
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2011-10-16 19
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2011-10-16 20
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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,
2011-10-16 21
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Determinant and Inverse of a Matrix（ 矩阵行列式和逆矩阵）,
? 1,Determinant of a matrix（ 矩阵行列式 ）,
2011-10-16 22
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? 2,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)
2011-10-16 23
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? 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 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 ）
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? 二阶逆矩阵运算,
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例,
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Eigenvalues 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 A
? ?
C? ( ) 0A I C???
?
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? 例,Find the eigenvalues and eigenvectors of
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2011-10-16 32
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2,Random Vector（ 随机向量）
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3,Multivariate Data Matrix(多元数据矩阵）
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4,mean vector,covariance and correlation matrices
（ 均值向量、协方差、相关系数矩阵）－总体
2011-10-16 35
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2011-10-16 36
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4,mean vector,covariance and correlation matrices
（ 均值向量、协方差、相关系数矩阵）－样本
2011-10-16 37
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5、欧氏距离与马氏距离
