Tenney–Euclidean tuning: Difference between revisions
Wikispaces>clumma **Imported revision 588204515 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 615647327 - Original comment: ** |
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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 | 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` | ||
# A`A and AA` are symmetric matrices | # A`A and AA` are symmetric matrices | ||
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<!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:4 -->The pseudoinverse</h1> | <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="The pseudoinverse"></a><!-- ws:end:WikiTextHeadingRule:4 -->The pseudoinverse</h1> | ||
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 /> | 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 /> | ||
<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` = A*(AA*)^(-1), where A* 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` = A*(AA*)^(-1), where A* 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 /> | ||