description/proof of that component of unitary matrix has absolute value equal to or smaller than \(1\)
Topics
About: matrices space
The table of contents of this article
Starting Context
- The reader knows a definition of unitary matrix.
Target Context
- The reader will have a description and a proof of the proposition that each component of any unitary matrix has an absolute value equal to or smaller than \(1\).
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(M\): \(\in \{\text{ the } n \times n \text{ unitary matrices }\}\)
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Statements:
\(\forall j, l \in \{1, ..., n\} (\vert M^j_l \vert \le 1)\)
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2: Proof
Whole Strategy: Step 1: see that \(\sum_{l \in \{1, ..., n\}} \vert M^j_l \vert^2 = 1\).
Step 1:
\(M M^* = I\), by the definition of unitary matrix.
Let \(j \in \{1, ..., n\}\) be any.
\((M M^*)^j_j = I^j_j = 1\), but he left hand side is \(\sum_{l \in \{1, ..., n\}} M^j_l {M^*}^l_j = \sum_{l \in \{1, ..., n\}} M^j_l \overline{M^j_l} = \sum_{l \in \{1, ..., n\}} \vert M^j_l \vert^2\).
So, for each \(l \in \{1, ..., n\}\), \(\vert M^j_l \vert^2 \le 1\).
So, \(\vert M^j_l \vert \le 1\) for each \(j, l \in \{1, ..., n\}\).
