Cauchy-Schwarz Inequality/Complex Numbers
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Theorem
- $\ds \paren {\sum \cmod {w_i}^2} \paren {\sum \cmod {z_i}^2} \ge \cmod {\sum w_i z_i}^2$
where all of $w_i, z_i \in \C$.
Proof
Let $w_1, w_2, \ldots, w_n$ and $z_1, z_2, \ldots, z_n$ be arbitrary complex numbers.
Take the Binet-Cauchy Identity:
- $\ds \paren {\sum_{i \mathop = 1}^n a_i c_i} \paren {\sum_{j \mathop = 1}^n b_j d_j} = \paren {\sum_{i \mathop = 1}^n a_i d_i} \paren {\sum_{j \mathop = 1}^n b_j c_j} + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {a_i b_j - a_j b_i} \paren {c_i d_j - c_j d_i}$
and set $a_i = w_i, b_j = \overline {z_j}, c_i = \overline {w_i}, d_j = z_j $.
This gives us:
\(\ds \paren {\sum_{i \mathop = 1}^n w_i \overline {w_i} } \paren {\sum_{j \mathop = 1}^n \overline {z_j} z_j}\) | \(=\) | \(\ds \paren {\sum_{i \mathop = 1}^n w_i z_i} \paren {\sum_{j \mathop = 1}^n \overline {z_j} \overline {w_j} }\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {w_i \overline {z_j} - w_j \overline {z_i} } \paren {\overline {w_i} z_j - \overline {w_j} z_i}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\sum_{i \mathop = 1}^n w_i \overline {w_i} } \paren {\sum_{j \mathop = 1}^n \overline {z_j} z_j}\) | \(=\) | \(\ds \paren {\sum_{i \mathop = 1}^n w_i z_i} \overline {\paren {\sum_{i \mathop = 1}^n w_i z_i} }\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \sum_{1 \mathop \le i \mathop < j \mathop \le n} \paren {w_i \overline {z_j} - w_j \overline {z_i} } \overline {\paren {w_i \overline {z_j} - w_j \overline {z_i} } }\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\sum_{i \mathop = 1}^n \cmod {w_i}^2} \paren {\sum_{j \mathop = 1}^n \cmod {z_j}^2}\) | \(=\) | \(\ds \cmod {\sum_{i \mathop = 1}^n w_i z_i}^2 + \sum_{1 \mathop \le i \mathop < j \mathop \le n} \cmod {w_i \overline {z_j} - w_j \overline {z_i} }^2\) | Modulus in Terms of Conjugate | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\sum_{i \mathop = 1}^n \cmod {w_i}^2} \paren {\sum_{j \mathop = 1}^n \cmod {z_j}^2}\) | \(\ge\) | \(\ds \cmod {\sum_{i \mathop = 1}^n w_i z_i}^2\) | Complex Modulus is Non-Negative |
Hence the result.
$\blacksquare$
Source of Name
This entry was named for Augustin Louis Cauchy and Karl Hermann Amandus Schwarz.
Sources
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $168$
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: Exercise $30$