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Statistical Analysis. Professor Lynne Stokes Department of Statistical Science Lecture 5QF Introduction to Vector and Matrix Operations Needed for the Theory of Quadratic Forms. One Factor (Fixed Effects) General Linear Model. Common Matrix Form. Regression: X full column rank
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Statistical Analysis Professor Lynne Stokes Department of Statistical Science Lecture 5QF Introduction to Vector and Matrix Operations Needed for the Theory of Quadratic Forms
One Factor (Fixed Effects) General Linear Model Common Matrix Form Regression: X full column rank GLM: X less than full column rank Linear Statistical Models Regression Model
Notation Response Vector Error Vector Design / Regressor Matrix General Matrices : A, B, …
Matrix Rank Linear Independence Can’t Express any of the Vectors as a Linear Combination of the Other Vectors Rank of a Matrix Maximum Number of Linearly Independent Columns (Row Rank = Column Rank) Note: A square matrix with a nonzero determinant is full rank, or nonsingular.
Special Matrices Diagonal Matrix Identity Matrix
Special Matrices Matrix of Ones (any dimensions) Null Matrix
Vector Multiplication Matrix Multiplication A must have the same number of columns as B has rows: A (n x s), B (s x k) Matrix Operations Addition A and B must have the same dimensions
Inverse Matrix A has an inverse, denoted A-1 if and only if (a) A is a square (n x n) matrix and (b) A is of full (row, column) rank. Then AA-1 = A-1A = I. A matrix inverse is unique. Matrix Operations Transpose Interchange rows and columns Symmetric Matrix: A (n x n) with A = A’ i.e, aij = aji
Orthonormal Matrix Symmetric Idempotent Matrix Note : then A-1 = A’ Only Full-Rank Symmetric Idempotent Matrix: I Note: A matrix all of whose columns are mutually orthogonal is called an orthogonal matrix. Often “orthogonal” is used in place of “orthonormal.” Special Vector and Matrix Properties Orthogonal Vectors Normalized Vectors a’b = 0
Eigenvectors: v1, v2, …, vn Eigenvalues and Eigenvectors A is square (n x n) and symmetric: All eigenvalues and eigenvectors are real-valued. Eigenvalues: l1, l2, …, ln (solve an nth degree polynomial equation in l) Note: If all eigenvalues are distinct, all eigenvectors are mutually orthogonal and can, without loss of generality, be normalized. If some eigenvalues have multiplicities greater than 1, the corresponding eigenvectors can be made to be orthogonal. Eigenvectors are unique up to a multiple of –1.
Eigenvalues and Rank • The rank of a symmetric matrix equals the number of nonzero eigenvalues • All the eigenvalues of an idempotent matrix are 0 or 1 • It’s rank equals the number of eigenvalues that are 1 • The sum of its diagonal elements equals its rank • A diagonal matrix has its eigenvalues equal to the diagonal elements of the matrix • The identity matrix has all its eigenvalues equal to 1 • Any set of mutually orthonormal vectors can be used as eigenvectors
Quadratic Forms A can always be assumed to be symmetric: For any B, x’Bx = x’Ax with aij = (bij + bji)/2
Assignment 3 • Determine the rank of each of these matrices. • For each full-rank matrix, find its inverse. • Determine whether any of these matrices are orthogonal • Determine whether any of these matrices are idempotent. • Find the eigenvalues and eigenvectors of A and B.
