Question

For which 2×2 matrices Ais

$\left(\overrightarrow{v},\overrightarrow{w}\right)\mathbf{=}{\overrightarrow{\mathbf{v}}}^{\mathbf{T}}\mathbf{A}\overrightarrow{\mathbf{w}}$ an inner product in ${\mathit{R}}^{\mathit{2}}$? Hint: Be prepared to complete a square.

Short answer

The given product is an inner product if a>0, b=c and$d>b^2$

Step-by-step solution

Inner product of the matrix.

To find the inner product of the matrix.

$A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],\stackrel{r}{v}=\left[\begin{array}{c}{v}_1\\ v_2\end{array}\right],\stackrel{r}{w}=\left[\begin{array}{c}{w}_1\\ w_2\end{array}\right]$

Let ,

Consider the following:

$$\begin{gathered} \left(\stackrel{r}{v},\stackrel{r}{w}\right)=\left[\begin{array}{cc}{v}_1& v_2\end{array}\right]\left[\begin{array}{ccc}a& b& w_1\\ c& d& w_2\end{array}\right] \\ =\left[\begin{array}{cc}{v}_1& v_2\end{array}\right]\left[\begin{array}{c}a{w}_1+b{w}_2\\ c{w}_1+d{w}_2\end{array}\right] \\ =a{w}_1{v}_1+b{w}_2{v}_2+c{w}_1{v}_2+d{w}_2{v}_2 \end{gathered}$$

Now, observe that if $\stackrel{r}{v}=\left[\begin{array}{c}1\\ 0\end{array}\right]=\stackrel{r}{w}$then,$\left(\stackrel{r}{v},\stackrel{r}{v}\right)=a>0$.

Thus, it is written as $\left(\stackrel{r}{v},\stackrel{r}{v}\right)=a\left(w_1{v}_1+\frac{b}{a}{w}_2{v}_2+\frac{c}{a}{w}_1{v}_2+\frac{d}{a}{v}_2{w}_2\right)$.

But from the exercise 15 it will be an inner product if and only if when $\frac{b}{a}=\frac{c}{a}$and $\frac{d}{a}>{\left(\frac{b}{a}\right)}^2$.

Therefore, a>0, b=a and $d>b^2$.

Hence, the given product will be an inner product if a>0, b=c and$d>b^2$.