# Let 1 n 1 n be the vector in R n R n whose components are all

## Question:

Let ${\mathbf{1}}_{n}$ be the vector in ${\mathbb{R}}^{n}$ whose components are all one. Show that ${J}_{n}=$ ${\mathbf{1}}_{n}{\mathbf{1}}_{n}^{\mathrm{\prime}}/n$ is a projection matrix, i.e., that ${J}_{n}^{2}={J}_{n}$, and that it has rank one: tr(J_{n}) = 1$\text{Extra left or missing right}$. Show also that ${I}_{n}-{J}_{n}$ is the complementary projection of rank $n-1$.

Each of the quadratic forms in Exercise 1.15 can be expressed in the form ${Y}^{\mathrm{\prime}}{M}_{r}Y$, where each ${M}_{r}$ is a projection matrix of order $mn\times mn$. Show that each matrix is a Kronecker product

Find the rank of each matrix.

**Exercise 1.15**

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## Step by Step Answer:

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