Exercise 6.7.2

For these problems, choose an orthonormal basis $\alpha$, usually the standard basis, for the inner product space. Write down $A={[T]}_{\alpha }$. Pick an orthonormal basis $\beta$ such that ${[T^{\text{∗}}T]}_{\beta }$ is diagonal. Here the order of $\beta$ should follow the order of the value of its eigenvalue. The positive square roots of eigenvalues of $A^{\text{∗}}A$ is the singular values of $A$ and $T$. Extend $T(\beta )$ to be an orthonormal basis $\gamma$ for $W$. Then we have

and

1.

Pick $\alpha$ to be the standard basis. We have

$$\beta =\left\{(1,0),(0,1)\right\},$$

$$\gamma =\left\{\frac{1}{\sqrt{3}}(1,1,1),\frac{1}{\sqrt{2}}(0,1,-1),\frac{\sqrt{8}}{\sqrt{3}}(-\frac{1}{2},\frac{1}{4},\frac{1}{4})\right\},$$

and the singular values are $\sqrt{3},\sqrt{2}$.

2.

Pick

$$\alpha =\left\{f_1=\frac{1}{\sqrt{2}},f_2=\sqrt{\frac{3}{2}}x,f_3=\sqrt{\frac{5}{8}}(3x^2-1)\right\}.$$

We have

$$\beta =\left\{f_3,f_1,f_2\right\},$$

$$\gamma =\left\{f_1,f_2\right\},$$

and the singular values are $\sqrt{45}$.

3.

Pick

$$\alpha =\left\{f_1=\frac{1}{\sqrt{2\pi }},f_2=\frac{1}{\sqrt{\pi }}\sin <!--\; FUNCTION\; APPLICATION\; -->x,f_3=\frac{1}{\sqrt{\pi }}\cos <!--\; FUNCTION\; APPLICATION\; -->x\right\}.$$

We have

$$\beta =\left\{f_2,f_3,f_1\right\},$$

$$\gamma =\left\{\frac{1}{\sqrt{5}}(2f_2+f_3),\frac{1}{\sqrt{5}}(2f_3-f_2),f_1\right\},$$

and the singular values are $\sqrt{5},\sqrt{5},\sqrt{4}$.

4.

Pick $\alpha$ to be he standard basis. We have

$$\beta =\left\{\frac{1}{\sqrt{3}}(1,i+1),\sqrt{\frac{2}{3}}(1,-\frac{i+1}{2})\right\},$$

$$\gamma =\left\{\frac{1}{\sqrt{3}}(1,i+1),\sqrt{\frac{2}{3}}(1,-\frac{i+1}{2})\right\},$$

and the singular values are $2,1$.