Linear Algebra/Complex Representations

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Linear Algebra
 ← Factoring and Complex Numbers: A Review Complex Representations Definition and Examples of Similarity → 

Recall the definitions of the complex number addition


(a+bi)\,+\,(c+di)=(a+c)+(b+d)i

and multiplication.

\begin{array}{rl}
(a+bi)(c+di) &=ac+adi+bci+bd(-1)  \\
&=(ac-bd)+(ad+bc)i
\end{array}
Example 2.1

For instance,  (1-2i)\,+\,(5+4i)=6+2i and  (2-3i)(4-0.5i)=6.5-13i .

Handling scalar operations with those rules, all of the operations that we've covered for real vector spaces carry over unchanged.

Example 2.2

Matrix multiplication is the same, although the scalar arithmetic involves more bookkeeping.


\begin{pmatrix}
1+1i  &2-0i  \\
i    &-2+3i
\end{pmatrix}
\begin{pmatrix}
1+0i  &1-0i  \\
3i    &-i
\end{pmatrix}

\begin{align}
&=\begin{pmatrix}
(1+1i)\cdot(1+0i)+(2-0i)\cdot(3i)   &(1+1i)\cdot(1-0i)+(2-0i)\cdot(-i) \\
(i)\cdot(1+0i)+(-2+3i)\cdot(3i)     &(i)\cdot(1-0i)+(-2+3i)\cdot(-i)
\end{pmatrix}                                                  \\
&=\begin{pmatrix}
1+7i  &1-1i  \\
-9-5i  &3+3i
\end{pmatrix}
\end{align}

Everything else from prior chapters that we can, we shall also carry over unchanged. For instance, we shall call this


\langle \begin{pmatrix} 1+0i \\ 0+0i \\ \vdots \\ 0+0i \end{pmatrix},
\dots,
\begin{pmatrix} 0+0i \\ 0+0i \\ \vdots \\ 1+0i \end{pmatrix},
\begin{pmatrix} 0+1i \\ 0+0i \\ \vdots \\ 0+0i \end{pmatrix},
\dots,
\begin{pmatrix} 0+0i \\ 0+0i \\ \vdots \\ 0+1i \end{pmatrix} \rangle

the standard basis for  \mathbb{C}^n as a vector space over \mathbb{C} and again denote it \mathcal{E}_n .

Linear Algebra
 ← Factoring and Complex Numbers: A Review Complex Representations Definition and Examples of Similarity →