Solving a complex equation $Ax=b$ with $A,b\in\mathbb C$ using linear algebra

Question

Consider the following very simple linear system with one unknown:

\begin{equation}\label{a}\tag{1} Ax=b \\ \left ( 3+4i \right )x=(6+8i). \end{equation}

This paper ("On the numerical solving of complex linear systems") says that I can solve the linear system by transforming A to matrix form and then solving it as follows:

\begin{equation}\label{b}\tag{2} \begin{pmatrix} 3 & -4\\ 4 & 3 \end{pmatrix} \binom{x_r}{x_c} = \binom{b_r}{b_c}, \end{equation} where $b_r = 6,b_c=8$.

Question: The translation of the A is fairly easy to understand. What I don't get is why b is not converted to matrix form yet solving the above system yields the correct answer. In other words, the following equations

\begin{equation}\label{c}\tag{3} \left\{\begin{matrix} 3x_r-4x_c = 6\\ 4x_r+3x_c = 8 \end{matrix}\right. \end{equation} makes no sense to me.

Note that, I know how to solve the linear system. I'm looking for a detailed explanation of what's happening between (1) and (2)

Thank you

Answer 1

Let's try to justify the equations given by (3). If we write out the product $(3+4i)x$, we have $$ (3+4i)x = (3 + 4i)(x_r + x_ci) = 3x_r + 3x_c i + 4x_r i + 4x_c i^2 \\ = [3x_r - 4x_c] + [4x_r + 3x_c]i $$ Now, in order for two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal. So, in order to have $(3+4i)x = b$, we must have $$ 3x_r - 4x_c = b_r\\ 4x_r + 3x_c = b_c $$ which is precisely the system of equations that you've come up with.

Answer 2

Strictly speaking they should be converted to matrix form and you would solve a block-matrix equation system.

$$\begin{bmatrix}3&-4\\4&3\end{bmatrix}\begin{bmatrix}x_r&-x_c\\x_c&x_r\end{bmatrix}=\begin{bmatrix}b_r&-b_c\\b_c&b_r\end{bmatrix}$$

It just happens that this will be the same thing for this example. For more advanced fields of numbers we will need to resort to this block-matrix embedding.

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As an example where the simplified version will fail we can take a look at permutations. A permutation among three different elements can be represented with binary matrices:

$$P_1 = \left[\begin{array}{ccc}0&1&0\\0&0&1\\1&0&0\end{array}\right], P_2 = \left[\begin{array}{ccc}0&1&0\\1&0&0\\0&0&1\end{array}\right]$$ Now let us say that if we apply $P_1$ to some permutation and it becomes $P_2$, then what was the original permutation?

This we can express as: $$P_1X = P_2$$

Here we need $X$ to be a 3x3 matrix, because there is no other way to represent it.

The answer is: $$X=P_1 ^{-1}P_2$$

Answer 3

$x=x_r+ix_i$. When we multiply with $3+4i$ we get $3x_r+3ix_i+4ix_r-4x_i=6+8i$. We now equate the real and imaginary parts separately:$$3x_r-4x_i=6\\3x_i+4x_r=8$$

Answer 4

You want to solve $az=b$ for $a,b\in\mathbb C$, using the machinery of linear algebra.

Denote real and imaginary parts of the quantities as $z=x+iy$ and $a=a_R + i a_I, b= b_R+i b_I$. The equation is then equivalent to the following system of real equations: $$\begin{cases} a_R x - a_I y = b_R, \\ a_R y + a_I x = b_I, \end{cases}$$ which can be written in matrix form as $$\underbrace{\begin{pmatrix} a_R & -a_I \\ a_I & a_R \end{pmatrix}}_{\equiv A} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} b_R \\ b_I \end{pmatrix}. $$ Note that the determinant of the matrix $A$ equals $|a|^2$, which is consistent with $az=b$ having nontrivial solutions iff $a\neq0$.

Assuming $|a|^2\neq0$, we can invert $A$ obtaining $$\begin{pmatrix} x \\ y \end{pmatrix} = \frac{1}{|a|^2}\begin{pmatrix} a_R & a_I \\ - a_I & a_R \end{pmatrix} \begin{pmatrix} b_R \\ b_I \end{pmatrix},$$ which translated back into a linear system is $$\begin{cases} |a|^2 x = a_R b_R + a_I b_I, \\ |a|^2 y = -a_I b_R + a_R b_I, \end{cases}$$ or summing the first equation to the second times $i$, gives $$z\equiv x+iy = \frac{1}{|a|^2}[(a_R b_R + a_I b_I) + i (-a_I b_R+a_R b_I)].$$ To see that this is consistent with the solution we get via complex analysis, $z=b/a$, write the latter as $z=ba^*/|a|^2$, and verify that the two expressions are equal.