- #1
kryptyk
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I've been looking into Geometric Algebra approaches to linear transformations and have found it to be MUCH nicer than the conventional matrix approaches for certain kinds of transformations. Moreover, I find it much more intuitive, particularly in its way of dealing with complex numbers.
For instance, consider some linear operator M from R^n to R^n. If all its eigenvalues are real, it is easy enough to see how it acts on linear subspaces. But how are we to geometrically interpret complex eigenvalues and their corresponding eigenvectors?
If n=2, this is relatively simple. We can treat complex eigenvalues as scalings and rotations on the plane. In fact, we can use the following isomorphism between C and 2 \times 2 antisymmetrical matrices over R \;:
a + i b \longleftrightarrow \left(\begin{array}{cc}a&-b\\b&a\end{array}\right)
But the use of eigenvectors with complex-valued coordinates can get quite ugly - especially when dealing with spaces of greater dimension than 2. Especially if we're only interested in how the operator acts on real vectors.
However, while we cannot generally identify rotations with real eigenvectors, we can identify rotations with real eigenbivectors where the eigenbivectors represent plane elements rather than line elements. The corresponding eigenvalues then express a scaling of areas rather than lengths. This then extends naturally to higher dimensions.
Moreover, GA provides a way to express any linear map as a geometric product without the use of any matrices. Certain kinds of maps, such as rotations and reflections, afford extremely simple representations in this way that not only more clearly illustrate the essence of the map but also are much less tedious to work with than matrices. These ideas are so extremely powerful I'm surprised they are seldom mentioned in the literature.
I was wondering if anyone here is familiar with these methods, and even if not, if anyone would be interested in looking further into these methods with me.
Thanks!
For instance, consider some linear operator M from R^n to R^n. If all its eigenvalues are real, it is easy enough to see how it acts on linear subspaces. But how are we to geometrically interpret complex eigenvalues and their corresponding eigenvectors?
If n=2, this is relatively simple. We can treat complex eigenvalues as scalings and rotations on the plane. In fact, we can use the following isomorphism between C and 2 \times 2 antisymmetrical matrices over R \;:
a + i b \longleftrightarrow \left(\begin{array}{cc}a&-b\\b&a\end{array}\right)
But the use of eigenvectors with complex-valued coordinates can get quite ugly - especially when dealing with spaces of greater dimension than 2. Especially if we're only interested in how the operator acts on real vectors.
However, while we cannot generally identify rotations with real eigenvectors, we can identify rotations with real eigenbivectors where the eigenbivectors represent plane elements rather than line elements. The corresponding eigenvalues then express a scaling of areas rather than lengths. This then extends naturally to higher dimensions.
Moreover, GA provides a way to express any linear map as a geometric product without the use of any matrices. Certain kinds of maps, such as rotations and reflections, afford extremely simple representations in this way that not only more clearly illustrate the essence of the map but also are much less tedious to work with than matrices. These ideas are so extremely powerful I'm surprised they are seldom mentioned in the literature.
I was wondering if anyone here is familiar with these methods, and even if not, if anyone would be interested in looking further into these methods with me.
Thanks!