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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 l.jpg
Statistical Analysis

Professor Lynne Stokes

Department of Statistical Science

Lecture 5QF

Introduction to Vector and Matrix Operations Needed for the Theory of Quadratic Forms

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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

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Response Vector

Error Vector

Design / Regressor Matrix

General Matrices : A, B, …

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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.

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Special Matrices

Diagonal Matrix

Identity Matrix

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Special Matrices

Matrix of Ones

(any dimensions)

Null Matrix

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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


A and B must have the same


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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


Interchange rows and columns

Symmetric Matrix: A (n x n) with A = A’ i.e, aij = aji

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Orthonormal Matrix

Symmetric Idempotent Matrix

Note : then A-1 = A’

Only Full-Rank Symmetric Idempotent Matrix:


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

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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.

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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

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Quadratic Forms

A can always be assumed to be symmetric:

For any B, x’Bx = x’Ax with aij = (bij + bji)/2

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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.