Question

Consider an$\mathit{n}\mathbf{\times }\mathit{m}$ matrix $\mathit{A}\mathbf{=}\mathit{Q}\mathit{R}\mathbf{,}$ where $\mathit{Q}$ is a $\mathit{n}\mathbf{\times }\mathit{m}$ matrix with Orthonormality columns and $\mathit{R}$is an upper triangular $\mathit{m}\mathbf{\times }\mathit{m}$ matrix with positive diagonal entries ${\mathit{r}}_{\mathbf{11}}\mathbf{,}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{,}{\mathit{r}}_{\mathbf{m}\mathbf{m}}$ . Express$\mathit{d}\mathit{e}\mathit{t}\left(A^{T}A\right)$ in terms of the scalars${\mathit{r}}_{\mathbf{i}\mathbf{i}}$ . What can you say about the sign of $\mathit{d}\mathit{e}\mathit{t}\left(A^{T}A\right)$?

Short answer

Therefore, the $\det \left(A^{T}A\right)$sign is always positive.

Step-by-step solution

Step by Step Solution: Step 1: Matrix Definition

Matrixis aset of numbers arranged inrowsandcolumnsso as to form a rectangular array.

The numbers are called the elements, or entries, of the matrix.

If there are $m$rows and $n$columns, the matrix is said to be a “ $m$by $n$” matrix, written “ $m\times n$.”

Step 2: To find  det(ATA) is positive or negative

To solve,

$$\begin{gathered} det\left(A^{T}A\right)=det\left({\left(QR\right)}^{T}QR\right) \\ det\left(A^{T}A\right)=det\left(R^T{Q}^{T}QR\right) \\ det\left(A^{T}A\right)=det\left(R^{T}R\right) \\ det\left(A^{T}A\right)=det{R}^{T}detR \\ det\left(A^{T}A\right)=detRdetR \\ det\left(A^{T}A\right)=(detR{)}^2 \end{gathered}$$

$\det \left(A^{T}A\right)={\left(\prod _{i=1}^m{r}_{i,i}\right)}^2>0$

Therefore, the $\det \left(A^{T}A\right)$ sign is always positive.