Extension of 3c to more than two groups. It is essentially a review of the requisite matrix tools and is not intended to be a complete development. However, it is sufficiently self-contained so that those with no previous exposure to the subject should need no other reference.
Anyone unfamiliar with matrix algebra should plan to work most of the problems entailing numerical illustrations. It would also be helpful to explore some of the problems involving general matrix manipulation. With the exception of a few derivations that seemed instructive, most of the results are given without proof. Some additional proofs are requested in the problems. For the remaining proofs, see any general text on matrix theory or one of the specialized matrix texts oriented to statistics, such as Graybill , Searle , or Harville We use uppercase boldface letters to represent matrices.
All entries in matrices will be real numbers or variables representing real numbers. The elements of a matrix are displayed in brackets. The matrix A in 2. With three rows and two columns, the matrix A in 2. A vector is a matrix with a single column or row. The following could be the test scores of a student in a course in multivariate analysis: 2. The transpose operation is defined in Section 2. Geometrically, a vector with p elements identifies a point in a p-dimensional space. The elements in the vector are the coordinates of the point.
In Section 3. In some cases, we will be interested in a directed line segment or arrow from the origin to the point. A single real number is called a scalar, to distinguish it from a vector or matrix. Thus two matrices of the same size are unequal if they differ in a single position. The following examples illustrate the transpose of a matrix or vector: The transpose operation does not change a scalar, since it has only one row and one column. If the transpose operator is applied twice to any matrix, the result is the original matrix: 2. For example, Clearly, all symmetric matrices are square.
For example, in the matrix the elements 5, 9, and 1 lie on the diagonal. If a matrix contains zeros in all off-diagonal positions, it is said to be a diagonal matrix. An example of a diagonal matrix is This matrix could also be denoted as 2. This is denoted by diag A. Thus for the above matrix A, we have 2. For example, 2. For example, Similarly, the difference between two matrices or two vectors of the same size is found by subtracting corresponding elements. For example, Matrix addition is commutative: 2. We therefore multiply each row of A by each column of B, and the size of AB consists of the number of rows of A and the number of columns of B.
For example, if then In this case, AB is of a different size than either A or B. Clearly, A2 is defined only if A is square. In some cases AB is defined, but BA is not defined.
Sometimes AB and BA are both defined but are different in size. However, 2. For example, let Then Thus we must be careful to specify the order of multiplication. Multiplication is distributive over addition or subtraction: 2. They need not be square. Multiplication involving vectors follows the same rules as for matrices.
For example, let Then Note thatA and B must be conformable for multiplication, and B and C must be conformable. We can sometimes factor a sum of triple products on both the right and left sides. Then 2. The square root of the sum of squares of the elements of a is the distance from the origin to the point a and is also referred to as the length of a: 2. For example, if A has three rows and B has two columns, then the product AB can be written as 2.
Thus AB can be written in the form This result holds in general: 2. Thus if we write A in terms of its rows as then we have 2. The product of a scalar and a matrix is obtained by multiplying each element of the matrix by the scalar: 2. Either of these is, of course, a scalar and can be treated as such. Expressions such as are permissible assuming A is positive definite; see Section 2. It can be seen that this formulation is equivalent to the usual row-by-column definition of matrix multiplication. For example, the 1, 1 element of AB is the product of the first row of A and the first column of B.
In the 1, 1 element of A11B11 we have the sum of products of part of the first row of A and part of the first column of B. In the 1, 1 element of A12B21 we have the sum of products of the rest of the first row of Aand the remainder of the first column of B.
Multiplication of a matrix and a vector can also be carried out in partitioned form. The partitioned multiplication in 2. For example, let Then Using a linear combination of columns of A as in 2. A set of vectors a1, a2, …, an is said to be linearly dependent if constants c1, c2, …, cn not all zero can be found such that 2. Thus linear dependence of a set of vectors implies redundancy in the set. Among linearly independent vectors there is no redundancy of this type. It can be shown that the number of linearly independent rows of a matrix is always equal to the number of linearly independent columns.
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For example, has rank 2 because the two rows are linearly independent neither row is a multiple of the other. However, even though A is full rank, the columns are linearly dependent because rank 2 implies there are only two linearly independent columns. Thus, by 2. This is a direct consequence of the linear dependence of the column vectors of A. Thus in a matrix equation, we cannot, in general, cancel matrices from both sides of the equation.
Methods of Multivariate Analysis Rencher
There are two exceptions to this rule. One involves a nonsingular matrix to be defined in Section 2. The other special case occurs when the expression holds for all possible values of the matrix common to both sides of the equation. Then the first column of A equals the first column of B. Note that rectangular matrices do not have inverses as in 2. If A and B are the same size and nonsingular, then the inverse of their product is the product of their inverses in reverse order, 2.
If a matrix is nonsingular, it can be canceled from both sides of an equation, provided it appears on the left right on both sides. The inverse of the transpose of a nonsingular matrix is given by the transpose of the inverse: 2. One way to obtain a positive definite matrix is as follows: 2. We give one method below in 2. A positive definite matrix A can be factored into 2.
Methods Of Multivariate Analysis Wiley Series In Probability And Statistics
One way to obtain T is the Cholesky decomposition, which can be carried out in the following steps. Then the elements of T are found as follows: For example, let Then by the Cholesky method, we obtain Each product contains one element from every row and every column, and 2. The factors in each product are written so that the column subscripts appear in order of magnitude and each product is then preceded by a plus or minus sign according to whether the number of inversions in the row subscripts is even or odd. An inversion occurs whenever a larger number precedes a smaller one.
The symbol n! For larger matrices, other methods are available for manual computation, but determinants are typically evaluated by computer. If the square matrix A is singular, its determinant is 0: 2. If A is near singular, then there exists a linear combination of the columns that is close to 0, and A is also close to 0. If A is nonsingular, its determinant is nonzero: 2. The trace is, of course, a scalar. For example, suppose Then The trace of the sum of two square matrices is the sum of the traces of the two matrices: 2.
For example, let Then From 2. The vector a can always be normalized by dividing by its length,. Thus 2. It is clear from 2. We illustrate the creation of an orthogonal matrix by starting with To normalize the three columns, we divide by the respective lengths, , , and , to obtain Note that the rows also became normalized and mutually orthogonal so that C satisfies both 2.
Alternatively, 2. If we multiply both sides of 2. To illustrate, we will find the eigenvalues and eigenvectors for the matrix The characteristic equation is In either case, x is the corresponding eigenvector. We illustrate 2. Using 2. The eigenvalues of a positive definite matrix are all positive. The eigenvalues of a positive semidefinite matrix are positive or zero, with the number of positive eigenvalues equal to the rank of the matrix.
The following result, known as the Perron—Frobenius theorem, is of interest in Chapter If all elements of the positive definite matrix A are positive, then all elements of the first eigenvector are positive. In this case, the nonzero eigenvalues of AB and BA will be the same. Therefore, by 2. Compare all 4 new copies. Book Description Wiley-Interscience. Condition: New. Seller Inventory NEW More information about this seller Contact this seller.
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The Second Edition contains revised and updated chapters from the critically acclaimed First Edition as well as brand-new chapters on:. Methods of Multivariate Analysis. Alvin C. Amstat News asked three review editors to rate their top five favorite books in the September issue. Methods of Multivariate Analysis was among those chosen.