19 Orthogonal projections and orthogonal matrices

... components, the cosine of the angle between x and y is computed with the same formula as in the standard basis, and so on. We can summarize this by saying that Euclidean geometry is invariant under orthogonal transformations. Exercise: ** Here’s another way to get at the same result. Suppose A is a ...

... components, the cosine of the angle between x and y is computed with the same formula as in the standard basis, and so on. We can summarize this by saying that Euclidean geometry is invariant under orthogonal transformations. Exercise: ** Here’s another way to get at the same result. Suppose A is a ...

13 Orthogonal groups

... on a vector space. Over the complex numbers there is essentially only one such form on a finite dimensional vector space, so we get the complex orthogonal groups On (C) of complex dimension n(n − 1)/2, whose Lie algebra is the skew symmetric matrices. Over the real numbers there are several differen ...

... on a vector space. Over the complex numbers there is essentially only one such form on a finite dimensional vector space, so we get the complex orthogonal groups On (C) of complex dimension n(n − 1)/2, whose Lie algebra is the skew symmetric matrices. Over the real numbers there are several differen ...