Your Textbook Shows That the Following Matrix (M_x = I_n $\frac { 1 } { n }$

question 16

Essay

Your textbook shows that the following matrix (M_x = I_n - P_x)is a symmetric idempotent matrix. Consider a different Matrix A, which is defined as follows: A = I - $\frac { 1 } { n }$ ιι' and ι = $\left[ \begin{array} { c } 1 \\1 \\\ldots \\1\end{array} \right]$ a. Show what the elements of A look like.
b. Show that A is a symmetric idempotent matrix
c. Show that Aι = 0.
d. Show that A $\hat { \mathrm { U } }$ = $\hat { \mathrm { U } }$ , where $\hat { \mathrm { U } }$ is the vector of OLS residuals from a multiple regression.

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Definitions:

Efferent

Referring to nerve fibers that carry impulses away from the central nervous system to effectors, such as muscles or glands.

Afferent

Pertaining to nerve fibers that carry sensory information towards the central nervous system.

Indifferent

Showing no particular interest or sympathy; apathetic or uninvolved emotionally.

Somatic Nervous System

A component of the peripheral nervous system that controls voluntary movements of the body through its control of skeletal muscles.