Consider the Following Symmetric and Idempotent Matrix A: a = $\frac { 1 } { n }$

question 9

Essay

Consider the following symmetric and idempotent Matrix A: A = I - $\frac { 1 } { n }$ ??' and ? = $\left[ \begin{array} { c } 1 \\1 \\\ldots \\1\end{array} \right]$ a. Show that by postmultiplying this matrix by the vector Y (the LHS variable of the OLS regression), you convert all observations of Y in deviations from the mean.
b. Derive the expression Y'AY. What is the order of this expression? Under what other name have you encountered this expression before?

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Definitions:

Functional Unity

The concept that all aspects of a society or system work together in an interrelated manner to ensure stability and continuity.

Robert Merton

An influential American sociologist known for developing theories on social structure, anomie, and for the concept of self-fulfilling prophecy, among other contributions to sociology.

Societal Community

A group of people who share common values, traditions, and behaviors, forming a cohesive unit within a larger society.

Integrative

Approaches or methods that combine different theories, practices, or systems into a unified whole.