Working with $z$, $\overline{z}$ instead of $\operatorname{Re}(z)$, $\operatorname{Im}(z)$

The problem is the following:

Determine conditions for $a,b,c\in\mathbb{C}$ such that $az+b\overline{z}+c=0$ has unique solution in $\mathbb{C}$.

Teacher answer:

If $az+b\overline{z}+c=0$, then $\overline{b}z+\overline{a}\overline{z}+\overline{c}=0$. So $\left(\begin{array}\\ a & b \\ \overline{b}&\overline{a} \end{array}\right)\left(\begin{array}\\ z \\ \overline{z} \end{array}\right)=\left(\begin{array}\\ -c \\ -\overline{c} \end{array}\right)$. Like in linear algebra, we have only one solution iff $|a|^2-|b|^2\neq 0$.

My answer is too long (equaling real and imaginary parts and taking a 2x2 system of linear equations), and teacher answer didn't convince me because teacher is solving a system where one component $z$ depends of another component which is $\overline{z}$ (if we obtain $z$ we automatically obtain $\overline{z}$ and viceversa).

Edit: Being more precise I have two conceptual questions, about validity and why the teacher's procedure works:

1) Why is solved for the pair $(z, \overline{z})$? if we obtain $z$, we also obtain its conjugate automatically. That has not sense.

2) Why do you need the conjugate of the first equation (is the same equation)?

3) Why condition $|a|^2-|b|^2\neq 0$ is necessary for unicity of solutions in the pair $z,\overline{z}$? I only know that there is unique solution for $(z,w)$ but we could have unique solution for $(z,\overline{z})$ and multiple solutions for $(z,w)$.

Solution 1:

1) The teacher's answer explains that there is only one solution for $x$, $y$ in

$$\begin{bmatrix}a&b\\\bar{b}&\bar{a}\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}-c\\-\bar{c}\end{bmatrix}$$

when $|a|^2-|b|^2\neq0$. It remains to see that that unique solution automatically has $y=\bar{x}$. Because if you actually solve for $x$ and $y$, you have

$$\begin{bmatrix}x\\y\end{bmatrix}=\frac{1}{|a|^2-|b|^2}\begin{bmatrix}\bar{a}&-b\\-\bar{b}&a\end{bmatrix}\begin{bmatrix}-c\\-\bar{c}\end{bmatrix}=\frac{1}{|a|^2-|b|^2}\begin{bmatrix}-\bar{a}c+b\bar{c}\\\bar{b}c-a\bar{c}\end{bmatrix} $$

and now it's clear that $x$ and $y$ are automatically conjugates.

2) The equation found from taking the conjugate is not "the same" equation, in a linear algebra sense. It is not a scalar multiple of the original equation, so in a linear algebra sense, it is a linearly independent equation.

3) If the determinant is $0$, then of course one possibility is that there are no solutions for $x$ and $y$. But there might be infinitely many solutions, and as you suggest, maybe exactly one of them also satisfies $y=\bar{x}$. I think you are right to worry about this when your teacher has not.

If there are infinitely many solutions, then the two equations' left-hand sides are linearly dependent, and we need only examine the first equation $ax+by=c$, leading us to $y=(c-ax)/b$ (assuming for now that $b\neq0$, where $b=0$ is a trivial case to consider). How could this work out to equal $\bar{x}$?

$$\begin{align}\bar{x}&=(c-ax)/b\\\implies x&=(\bar{c}-\bar{a}\bar{x})/\bar{b}\\&=(\bar{c}-\bar{a}(c-ax)/b)/\bar{b}\\\implies \bar{b}bx&=b\bar{c}-c\bar{a}+a\bar{a}x\\\implies \bar{b}bx-a\bar{a}x&=b\bar{c}-c\bar{a}\\\implies (|b|^2-|a|^2)x&=b\bar{c}-c\bar{a}\\\implies 0&=b\bar{c}-c\bar{a}\end{align}$$

So $b\bar{c}=c\bar{a}$. But then $$\begin{align}ax+b\bar{x}&=-c\\\implies a\bar{c}x+b\bar{c}\bar{x}&=-c\bar{c}\\\implies a\bar{c}x-\bar{a}c\bar{x}&=-|c|^2\\\implies a\bar{c}x-\overline{a\bar{c}x}&\phantom{=}\text{is a real number}\end{align}$$

This is only possible if in fact $c=0$. Then the equation becomes $ax+by=0$, a homogeneous equation. The solution vectors $(x,y)$ are all scalar multiples of each other, and so if one solution is indeed a pair of the form $(x,\bar{x})$, then all real multiples of that solution will be too.

Solution 2:

$(1)$ and $(2)$: Certainly, $\bar{z}$ is a function of $z$, however, $z$ and $\bar{z}$ are linearly independent. That is, there are no $a$ and $b$ so that $az+b\bar{z}=0$ for all $z\in\mathbb{C}$. However, we know that $\mathrm{Re}(z)=\frac12(z+\bar{z})$ and $\mathrm{Im}(z)=\frac1{2i}(z-\bar{z})$, and that lets us compute both the real and complex parts of $z$ as a linear combination of $z$ and $\bar{z}$. So, although $az+b\bar{z}+c=0\iff \bar{a}\bar{z}+\bar{b}z+\bar{c}=0$, in order to solve $az+b\bar{z}+c=0$ as a linear equation, we employ $\bar{a}\bar{z}+\bar{b}z+\bar{c}=0$.

$(3)$ Perhaps this might help to explain why these two equations relate to a real matrix equation. Note that $$ \begin{bmatrix} {\small1/2}&{\small1/2}\\ {\small-i/2}&{\small i/2} \end{bmatrix} \begin{bmatrix} z\\ \bar{z} \end{bmatrix} = \begin{bmatrix} z_r\\ z_i \end{bmatrix} $$ Here is the equation from your instructor $$ \begin{bmatrix} a&b\\ \bar{b}&\bar{a} \end{bmatrix} \begin{bmatrix} z\\ \bar{z} \end{bmatrix} = \begin{bmatrix} -c\\ -\bar{c} \end{bmatrix} $$ Multiply $\begin{bmatrix}{\small1/2}&{\small1/2}\\{\small-i/2}&{\small i/2}\end{bmatrix}$ on the left and insert the identity $$ \begin{align} \begin{bmatrix} {\small1/2}&{\small1/2}\\ {\small-i/2}&{\small i/2} \end{bmatrix} \begin{bmatrix} a&b\\ \bar{b}&\bar{a} \end{bmatrix} \overbrace{ \begin{bmatrix} 1&i\\ 1&-i \end{bmatrix} \begin{bmatrix} {\small1/2}&{\small1/2}\\ {\small-i/2}&{\small i/2} \end{bmatrix} }^\text{identity} \begin{bmatrix} z\\ \bar{z} \end{bmatrix} &= \begin{bmatrix} {\small1/2}&{\small1/2}\\ {\small-i/2}&{\small i/2} \end{bmatrix} \begin{bmatrix} -c\\ -\bar{c} \end{bmatrix}\\ \begin{bmatrix} {\small1/2}&{\small1/2}\\ {\small-i/2}&{\small i/2} \end{bmatrix} \begin{bmatrix} a+b&i(a-b)\\ \overline{a+b}&\overline{i(a-b)} \end{bmatrix} \begin{bmatrix} z_r\\ z_i \end{bmatrix} &= \begin{bmatrix} -c_r\\ -c_i \end{bmatrix}\\ \begin{bmatrix} a_r+b_r&b_i-a_i\\ a_i+b_i&a_r-b_r \end{bmatrix} \begin{bmatrix} z_r\\ z_i \end{bmatrix} &= \begin{bmatrix} -c_r\\ -c_i \end{bmatrix} \end{align} $$ The determinant of the last matrix is $(a_r^2+a_i^2)-(b_r^2+b_i^2)=|a|^2-|b|^2$.

Solution 3:

Consider the following fundamental property of complex numbers. If $z = x+i y$, where $x,y\in\mathbb{R}$, i.e., $$z = x+i y, \quad \bar z = x-i y,$$ then $$x = \frac{1}{2}(z+\bar z), \quad y = \frac{1}{2i}(z-\bar z).$$ That is, if we know $z,\bar z$ we can find $x,y$ uniquely and vice versa. Thus, it doesn't matter which way we solve the system. We can write it in terms of $x,y$ and solve for $x,y$ or write it in terms of $z,\bar z$ and solve for $z,\bar z$. For this particular problem, even if we're interested in $x,y$ it is easiest to solve for $z,\bar z$ and then find $x,y$.

A reader may claim we need only $z$ to find $x$ and $y$. Indeed, $x=\mathrm{Re}(z)$ and $y=\mathrm{Im}(z)$. But these operations are not linear. (For example, $\mathrm{Re}(i z) \ne i\mathrm{Re}(z)$.) To take advantage of linear algebra we must relate $x,y$ to $z,\overline z$ via linear transformations. These transformations are given above and in many of the other answers. Since the transformation is invertible, it is appropriate to treat $z,\overline z$ as independent, just as $x,y$ are independent. (This is made very explicit in @robjohn's answer.) Thus, starting from a linear equation in $z,\overline z$ we conjugate and treat the resulting two-by-two system as if $z,\overline z$ are independent. Of course, the fact that $z,\overline z$ are related by conjugation will restrict the space of solutions to a subspace of the space of solutions to the more general $z,w$ system.

This method is well known and often used in physics.

To answer your questions:

(1) If indeed we could solve the equation $az+b\bar z=c=0$ for $z$ directly, we would not need to introduce the conjugate equation. Since this is not possible we introduce the conjugate equation, arriving at a two-by-two system that we can solve.

(2) See the answer to (1). When thinking in terms of $z,\bar z$ we should consider the equation and its conjugate as "different equations" not "the same equation."

(3) The condition $$\mathrm{det}\left(\begin{array}{cc}a&b\\ \bar b&\bar a\end{array}\right) = |a|^2-|b^2|\ne 0$$ is the condition that the matrix $\left(\begin{array}{cc}a&b\\ \bar b&\bar a\end{array}\right)$ be invertible. If the matrix is not invertible the equation may have no solution or an infinite number of solutions. For example:

(a) The equation $z+\bar z-2i = 0$ has no solution.

(b) The equation $z+\bar z-2 = 0$ has an infinite number of solutions, $z = 1+i y$ where $y\in\mathbb{R}$.

Solution 4:

You know you can set this up as a real linear equation by splitting the complex variable $z$ into two real variables $x + iy$, and splitting the one complex equation into its real and imaginary parts to get two real equations.

So, we have a real system of equations $T\vec{v} = \vec{w}$. We know that the solution space is either:

• Empty
• Has a unique solution
• Has a one-dimensional family of solutions $\vec{v}_0 + t \vec{u}$ for $t \in \mathbb{R}$
• Has a two-dimensional family of solutions: i.e. every $\vec{v}$ is a solution.

We can take the same equation $T \vec{v} = \vec{w}$ and look for its complex solutions. Based on how solving linear equations works, it's clear the complex solution space will be of the same form as the real solution space, just with complex variables allowed. (e.g. in the third case above, we allow $t \in \mathbb{C}$ but keep the same $\vec{v}_0$ and $\vec{u}$)

But now that we've moved to complex linear algebra, we have a bit more freedom with our scalars! We can, for example, make the invertible change of basis:

$$ \left( \begin{matrix} 1 & i \\ 1 & -i \end{matrix} \right) \cdot \left( \begin{matrix}x \\ y \end{matrix} \right) = \left( \begin{matrix} z \\ \bar{z} \end{matrix} \right)$$

The use of the names $z$ and $\bar{z}$ are somewhat less than literal, since $x$ and $y$ are now allowed to be complex numbers.

Our complex system of equations have to be equivalent to the original complex equation and its conjugate (e.g. it comes from making substitutions like $z = x+iy$ and combining the equation and its conjugate linearly. Remember $x,y$ are still allowed to be complex!), which is the system of equations the teacher set up. But since this system of equations is just the complexification of a real system of equations in disguise, the complex solution space has to be of the same form as the real solution space, as described above, and so we can use the teacher's argument.

If one didn't believe things had to work out, you could explicitly compute the change of basis on the matrix:

$$ \left( \begin{matrix} 1 & i \\ 1 & -i \end{matrix} \right) \cdot \left( \begin{matrix} \Re a + \Re b & -\Im a + \Im b \\ \Im a + \Im b & \Re a - \Re b \end{matrix} \right) \cdot \left( \begin{matrix} 1 & i \\ 1 & -i \end{matrix} \right)^{-1} \\= \left( \begin{matrix} a + b & ia - ib \\ \bar{a} + \bar{b} & -i\bar{a} + i\bar{b} \end{matrix} \right) \cdot \left( \begin{matrix} 1 & 1 \\ -i & i \end{matrix} \right) \cdot \frac{1}{2} = \left( \begin{matrix} a & b \\ \bar{b} & \bar{a} \end{matrix} \right) $$