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The term <q>inner product</q> means slightly different things to different people. For some, it is the <q>dot product</q> that you may have seen in a calculus or physics course. Our inner product could be called the <q>Hermitian inner product</q> to emphasize the use of vectors over the complex numbers and conjugating some of the entries. So Sage has a <code>.dot_product()</code>, <code>.inner_product()</code>, and <code>.hermitian_inner_product()</code> <mdash /> we want to use the last one.<br /><br />
-Furthermore, Sage defines the Hermitian inner product by conjugating entries from the <em>first</em> vector, rather than the <em>second</em> vector as defined in the text. This is not as big a problem as it might seem. First, <acroref type="theorem" acro="IPAC" />, tells us that we can counteract a difference in order by just taking the conjugate, so you can translate between Sage and the text by taking conjugates of results achieved from a Hermitian inner product. Second, we will mostly be interested in when a Hermitian inner product is zero, which is its own conjugate, so no adjustment is required. Third, most theorems are true as stated with either definition, although there are exceptions, like <acroref type="theorem" acro="IPSM" />.<br /><br />
From now on, when we mention an inner product in the context of using Sage, we will mean <code>.hermitian_inner_product()</code>. We will redo the first part of <acroref type="example" acro="CSIP" />. Notice that the syntax is a bit asymmetric.
Norms are as easy as conjugates. Easier maybe. It might be useful to realize that Sage uses entirely distinct code to compute an exact norm over <code>QQbar</code> versus an approximate norm over <code>CDF</code>, though that is totally transparent as you issue commands. Here is <acroref type="example" acro="CNSV" /> reprised.