Tenney–Euclidean tuning: Difference between revisions
Wikispaces>genewardsmith **Imported revision 154244439 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 154255683 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-27 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-07-27 17:23:48 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>154255683</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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# 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 | ||
# A`A and AA` are | # A`A and AA` are symmetric matricies | ||
From these properties it can be deduced that | From these properties it can be deduced that | ||
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<!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-RMS tuning-The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:2 -->The pseudoinverse</h3> | <!-- ws:start:WikiTextHeadingRule:2:&lt;h3&gt; --><h3 id="toc1"><a name="x-RMS tuning-The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:2 -->The pseudoinverse</h3> | ||
If A is an mxn matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the nxm matrix such that<br /> | If A is an mxn matrix with real entries, and if we denote the pseudoinverse by A`, then it is defined as the nxm 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 | <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 matricies</li></ol><br /> | ||
From these properties it can be deduced that<br /> | From these properties it can be deduced that<br /> | ||
<ul><li>If A is invertible, its inverse is A`</li><li>If A has rational entries, so does A`</li><li>A`` = A</li><li>The pseudoinverse of the transpose is the transpose of the pseudoinverse</li><li>AA` is the orthogonal projection map onto the space spanned by the columns of A</li><li>A`A is the orthogonal projection map onto the space spanned by the rows of A</li><li>I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A</li><li>If the rows of A are linearly independent, then A` = At(AAt)^(-1), where At is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.</li><li>uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.</li></ul><br /> | <ul><li>If A is invertible, its inverse is A`</li><li>If A has rational entries, so does A`</li><li>A`` = A</li><li>The pseudoinverse of the transpose is the transpose of the pseudoinverse</li><li>AA` is the orthogonal projection map onto the space spanned by the columns of A</li><li>A`A is the orthogonal projection map onto the space spanned by the rows of A</li><li>I - A`A, where I is the identity matrix, is the orthogonal projection map onto the kernel, or null space, of A</li><li>If the rows of A are linearly independent, then A` = At(AAt)^(-1), where At is the transpose of A. This means the pseudoinverse can be found in this important special case by people who don't have a pseudoinverse routine available by using a matrix inverse routine.</li><li>uA` is the nearest point to u in the subspace spanned by the rows of A; A`v is the nearest point to v in the space spanned by the columns of A.</li></ul><br /> | ||