I'm trying to show that given a set $\{\mathbf{a}, \mathbf{b}\}$ of orthonormal vectors in a 2-dimensional vector space, I can construct the identity matrix by computing $aa^\dagger + bb^\dagger$. This should be straightforward but it's not working out. I get that my conditions for orthonormality are $$|a_1|^2+|a_2|^2 = 1,$$ $$|b_1|^2 + |b_2|^2 = 1,$$ $$a_1^*b_1+a_2^*b_2 = 0$$ but these don't directly lead me to the identity matrix. Where am I going wrong?
Orthogonal vectors. Where am I going wrong?
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vector-spaces
orthonormal
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2If you are over $\mathbb C$, shouldn't it read $aa^* + bb^*$? – 2012-09-21
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0Is your vector space complex? Otherwise the stars in your last equations don't seem to make sense. But if it _is_ complex, then the transposes in $aa^T+bb^T$ should rather be adjoints. – 2012-09-21
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1@MTurgeon: From context it seems clear that the vectors are rows. martini's point is that complex conjugates should be used if working over $\mathbb C$, which seems consistent with the last equation (where $a_1^*$ and $a_2^*$ appear). – 2012-09-21
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0I am using the convention that vectors are columns and this is a complex vector space. I'm also letting my components be $a_1$ and $a_2$ be components for $\mathbf{a}$ etc – 2012-09-21
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0@JonasMeyer I see. I completely misunderstood the question then. – 2012-09-21