QII
3x3 Matrices
Matrix Algebraic Properties and Abstract Reasoning
View
Answer the following questions concerning the curved surface given by Equation (3) in the Cartesian coordinate system $xyz$. Note that $\boldsymbol{m}^{\mathrm{T}}$ indicates transpose of $\boldsymbol{m}$.
$$f(x, y, z) = 2\left(x^{2} + y^{2} + z^{2}\right) + 4yz + \frac{z - y}{\sqrt{2}} = 0 \tag{3}$$
1. When the function $f(x, y, z)$ is expressed in the following form, derive the real symmetric matrix $\boldsymbol{A}$ of order 3 and the vector $\boldsymbol{b} = \left(\begin{array}{l} b_{1} \\ b_{2} \\ b_{3} \end{array}\right)$:
$$f(x, y, z) = \left(\begin{array}{lll} x & y & z \end{array}\right) \boldsymbol{A} \left(\begin{array}{l} x \\ y \\ z \end{array}\right) + 2\boldsymbol{b}^{\mathrm{T}} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$$
2. Suppose that the matrix $\boldsymbol{A}$ derived in Question II.1 is diagonalized as $\boldsymbol{A} = \boldsymbol{P}^{\mathrm{T}}\boldsymbol{D}\boldsymbol{P}$ using an orthogonal matrix $\boldsymbol{P}$ of order 3 and a diagonal matrix $\boldsymbol{D}$, which is given by Equation (5):
$$\boldsymbol{D} = \left(\begin{array}{ccc} d_{1} & 0 & 0 \\ 0 & d_{2} & 0 \\ 0 & 0 & d_{3} \end{array}\right) \tag{5}$$
Obtain a set of $\boldsymbol{P}$ and $\boldsymbol{D}$ satisfying $d_{1} \geq d_{2} \geq d_{3}$.
3. Express the function $f$ using $X$, $Y$, and $Z$, obtained by applying the coordinate transformation defined by $\left(\begin{array}{l} X \\ Y \\ Z \end{array}\right) = \boldsymbol{P} \left(\begin{array}{l} x \\ y \\ z \end{array}\right)$, using $\boldsymbol{P}$ derived in Question II.2.
4. Consider a region surrounded by the curved surface given by Equation (3) and a plane defined by $y - z - \sqrt{2} = 0$. Obtain the volume of this region.