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Understanding Partitioned Matrices and Their Inverses in Statistics

This document provides an in-depth exploration of partitioned matrices, detailing expressions to compute the inverse and determinant of an m x m matrix partitioned into 2 x 2 blocks. It includes examples relevant to regression modeling and the properties of nonnegative and positive matrices. Additionally, the document covers the importance of the vec operator in transforming matrices into vectors for statistical applications, emphasizing the distribution of the sample covariance and correlation matrices.

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Understanding Partitioned Matrices and Their Inverses in Statistics

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  1. Matriks OperatorPertemuan 9 Matakuliah : Matrix Algebra for Statistics Tahun : 2009

  2. Partitioned Matrices Purpose: To obtain expressions for the inverse and determinant of an m x m matrix A that is partitioned into the 2 x 2 block where A11is m1xm1. A12 is m1xm2. A21 is m2xml, and A22is m2xm2

  3. Contoh: Let Amxm and Bmxm partitioned as A, A11, and A22 are nonsingular matrices B = A-1 and partition B as

  4. Then, • B11 =(A11-A12A22-1A21)-1 =A11 +A11A12B22A21A11, b) B22= (A22– A21-1A12)-1 = A22-1 + A22-1 A21B11A12A22-1 c) B12= -A11-1A12B22, d) B21 =-A22-1A21 B11

  5. Matrix equation Yields, A11B11 +A12B21= Im1 A21B12 +A22B22= 1m2 A11B12 +A12B22= (0) A21B11 + A22B21= (0)

  6. Example 1 Consider the regression model y = Xᵦ + є, where y is Nx1, X is Nx(k + 1), β is (k + 1) x1, and є is N x 1. Suppose that β and X are partitioned as β= (β 1T, β 2T)T and X = (X1, X2) so that the product X1β1is defined

  7. Example 2 Let the m x m matrix A be partitioned If A22= Im2 and A12 = (0) or A21= (0), then IAI = IA11I· To find the determinant use the cofactor expansion formula for a determinant

  8. where B is the (m2- 1) x m1 matrix obtained by deleting the last row from A21 Repeating this process another (m2- 1) times yields IA I = IA11l. In a similar way we obtain IA I = IA11I. when A21 = (0). by repeatedly expanding along the last row.

  9. Nonnegatif Vector An m x n matrix A is a nonnegative matrix, indicated by A ≥ (0), if each element of A is nonnegative. Similarly, A is a positive matrix, indicated by A > (0), if each element of A is positive. We will write A ≥ B and A> B to mean that A-B ≥ (0) and A-B > (0), respectively.

  10. Any matrix A can be transformed to a nonnegative matrix by replacing each of its elements by its absolute value. This will be denoted by abs(A); that is, if A is an mxn matrix, then abs(A) is also an mxn matrix with (i,j)th element given by Ia ij I.

  11. Let A be an mxm matrix and x be an mx1 vector. If A ≥(0) and x > 0, then with similar inequalities holding when minimizing and maximizing over columns instead of rows

  12. Theorm 1 Let A be an m x m positive matrix. Then ρ(A) is positive and is an eigenvalue of A. In addition, there exists a positive eigenvector of A corresponding to the eigenvalue ρ(A).

  13. g Theorm 2 Let A be an mxm positive matrix and suppose that  is an eigenvalue of A satisfying = ρ(A). If x is any eigenvector corresponding to , then A abs(x) = ρ(A)abs(x)

  14. Theorm 3 If A is an mxm positive matrix, then the dimension of the eigenspace corresponding to the eigenvalue ρ(A) is one. Further, if A is an eigenvalue of A and Aρ(A), then II < ρ(A).

  15. VEC OPERATOR There are situations in which it is useful to transform a matrix to a vector that has as its elements the elements of the matrix. One such situation in statistics involves the study of the distribution of the sample covariance matrix S The operator that transforms a matrix to a vector is known as the vec operator

  16. If the mxn matrix A has aias its i th column, then vec(A) is the mnx1 vector given by

  17. Example A is 2x3 matrix, If a is mx1 and b is nx1, then abTis mxn and vec(abT) = vec([b1a, b2a, ... , bna]) =

  18. Theorm Let a and b be any two vectors, while A and B are two matrices of the same size. Then a) vec(a) = vec(aT) = a, b) vec(αbT) = b a, c) vec( αA + (βB) = α vec(A) + β vec(B), where α and β are scalars.

  19. Example Suppose that we are interested in the distribution of the sample covariance matrix or the distribution of the sample correlation matrix computed from a sample of observations on three different variables. The resulting sample covariance and correlation matrices would be of the form

  20. So that vec(S) = (S11 S12, S13, S12, S22, S23, S13, S23, S33)T, vec(R) = (1, rl2, r13, r12, 1, r23, r13, r23, 1)T Since both S and R are symmetric, there are redundant elements in vec(S) and vec(R). The elimination of these results in v(S) and v(R) given by v(S) = (S11, S12, S13, S22, S23, S33)T, v(R) = (I, rl2, r13, 1, r23, 1)T

  21. Eliminating the nonrandom 1 s from v(R), we obtain which contains all of the random variables in R.

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