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Chapter 9: Problem 10

Suppose A is a real symmetric matrix. If the eigenvalues of \(\mathbf{A}\) are only \(0 \mathrm{~s}\) and 1 s then prove that \(A\) is idempotent.

Short Answer

Expert verified
Given only 0s and 1s as eigenvalues aligns with the characteristics of an idempotent matrix, this proves that the real symmetric matrix \( \mathbf{A} \) is idempotent.

Step by step solution

01

Characteristics of an Idempotent Matrix

We start by understanding the characteristics of an idempotent matrix. An idempotent matrix is a matrix which, when squared (i.e., when it is multiplied by itself), results in itself. In other words, if \( \mathbf{A} \) is idempotent, then \( \mathbf{A}^2 = \mathbf{A} \).
02

Characteristics of Eigenvalues

The eigenvalues of a matrix are the roots of the characteristic equation, which is \( \text{det}(\mathbf{A} - \lambda \mathbf{I}) = 0 \). For an idempotent matrix, the eigenvalues will be 0s and 1s only. This is because the idempotent matrix property states that \( \mathbf{A}^2 = \mathbf{A} \), and thus the characteristic equation becomes \( (1 - \lambda)\lambda = 0 \). This equation only has roots at 0 and 1, proving that the eigenvalues for an idempotent matrix are 0s and 1s.
03

Proving the Statement

Given that all the eigenvalues of matrix \( \mathbf{A} \) are 0s and 1s only, this aligns with the characteristics of an idempotent matrix. It is thus proven that the real symmetric matrix \( \mathbf{A} \), with eigenvalues as 0s and 1s, is idempotent.

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Most popular questions from this chapter

Let \(A_{1}, A_{2}, \ldots, A_{k}\) be the matrices of \(k>2\) quadratic forms \(Q_{1}, Q_{2}, \ldots, Q_{k}\) in the observations of a random sample of size \(n\) from a distribution that is \(N\left(0, \sigma^{2}\right)\). Prove that the pairwise independence of these forms implies that they are mutually independent. Hint: \(\quad\) Show that \(\boldsymbol{A}_{i} \boldsymbol{A}_{j}=\mathbf{0}, i \neq j\), permits \(E\left[\exp \left(t_{1} Q_{1}+t_{2} Q_{2}+\cdots+t_{k} Q_{k}\right)\right]\) to be written as a product of the mgfs of \(Q_{1}, Q_{2}, \ldots, Q_{k}\).

Suppose \(\mathbf{X}\) is an \(n \times p\) matrix with rank \(p\). (a) Show that \(\operatorname{ker}\left(\mathbf{X}^{\prime} \mathbf{X}\right)=\operatorname{ker}(\mathbf{X})\). (b) Use part (a) and the last exercise to show that if \(\mathbf{X}\) has full column rank, then \(\mathbf{X}^{\prime} \mathbf{X}\) is nonsingular.

For the two-way interaction model, \((9.5 .15)\), show that the following decomposition of sums of squares is true: $$ \begin{aligned} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{\ldots}\right)^{2}=& b c \sum_{i=1}^{a}\left(\bar{X}_{i . .}-\bar{X}_{\ldots .}\right)^{2}+a c \sum_{j=1}^{b}\left(\bar{X}_{. j .}-\bar{X}_{\ldots}\right)^{2} \\ &+c \sum_{i=1}^{a} \sum_{j=1}^{b}\left(\bar{X}_{i j .}-\bar{X}_{i . .}-\bar{X}_{. j .}+\bar{X}_{\ldots}\right)^{2} \\ &+\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{c}\left(X_{i j k}-\bar{X}_{i j .}\right)^{2} \end{aligned} $$ that is, the total sum of squares is decomposed into that due to row differences, that due to column differences, that due to interaction, and that within cells.

Let \(Q=X_{1} X_{2}-X_{3} X_{4}\), where \(X_{1}, X_{2}, X_{3}, X_{4}\) is a random sample of size 4 from a distribution that is \(N\left(0, \sigma^{2}\right)\). Show that \(Q / \sigma^{2}\) does not have a chi-square distribution. Find the mgf of \(Q / \sigma^{2}\).

Using the notation of this section, assume that the means satisfy the condition that \(\mu=\mu_{1}+(b-1) d=\mu_{2}-d=\mu_{3}-d=\cdots=\mu_{b}-d .\) That is, the last \(b-1\) means are equal but differ from the first mean \(\mu_{1}\), provided that \(d \neq 0\). Let independent random samples of size \(a\) be taken from the \(b\) normal distributions with common unknown variance \(\sigma^{2}\). (a) Show that the maximum likelihood estimators of \(\mu\) and \(d\) are \(\hat{\mu}=\bar{X}_{. .}\) and $$ \hat{d}=\frac{\sum_{j=2}^{b} \bar{X}_{. j} /(b-1)-\bar{X}_{.1}}{b} $$ (b) Using Exercise \(9.2 .4\), find \(Q_{6}\) and \(Q_{7}=c \hat{d}^{2}\) so that, when \(d=0, Q_{7} / \sigma^{2}\) is \(\chi^{2}(1)\) and $$ \sum_{i=1}^{a} \sum_{j=1}^{b}\left(X_{i j}-\bar{X}_{. .}\right)^{2}=Q_{3}+Q_{6}+Q_{7} $$ (c) Argue that the three terms in the right-hand member of part (b), once divided by \(\sigma^{2}\), are independent random variables with chi-square distributions, provided that \(d=0\). (d) The ratio \(Q_{7} /\left(Q_{3}+Q_{6}\right)\) times what constant has an \(F\) -distribution, provided that \(d=0\) ? Note that this \(F\) is really the square of the two-sample \(T\) used to test the equality of the mean of the first distribution and the common mean of the other distributions, in which the last \(b-1\) samples are combined into one.
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