Tenney–Euclidean tuning: Difference between revisions
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This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
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===The pseudoinverse=== | ===The pseudoinverse=== | ||
If A is an | If A is an m×n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n×m matrix such that | ||
# AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself. | # AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself. | ||
# A`AA` = A | # A`AA` = A | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-TE tuning-The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:4 -->The pseudoinverse</h3> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h3&gt; --><h3 id="toc2"><a name="x-TE tuning-The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:4 -->The pseudoinverse</h3> | ||
If A is an | If A is an m×n matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the n×m matrix such that<br /> | ||
<ol><li>AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.</li><li>A`AA` = A</li><li>A`A and AA` are symmetric matrices</li></ol><br /> | <ol><li>AA`A = A. Hence, AA` maps the rows of A to itself and A`A the columns of A to itself.</li><li>A`AA` = A</li><li>A`A and AA` are symmetric matrices</li></ol><br /> | ||
From these properties it can be deduced that<br /> | From these properties it can be deduced that<br /> | ||