Overview
When the vectors are interpreted as column vectors and forms as row vectors, (that is, they are members of a finite dimensional vector space, with inner product equivalent to the dot product) the following relationships can be derived for the derivative of the inner product. Here {% |a\rangle %} is a column vector, {% \langle a | %} is a row vector, and {% M %} is a matrix.
Definition
{% F = \alpha \langle a | b \rangle
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\frac{dF}{d \langle a |} = \alpha | b \rangle
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\frac{dF}{d | a \rangle} = \alpha \langle b | %}
{% F = \alpha \langle a | a \rangle
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\frac{dF}{d \langle a |} = 2 \alpha | a \rangle
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\frac{dF}{d | a \rangle} = 2 \alpha \langle a |
%}
{% F = \alpha \langle a | M | b \rangle
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\frac{dF}{d \langle a |} = \alpha M | a \rangle
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\frac{dF}{d | a \rangle} = \alpha \langle b | M^T
%}
{% F = \alpha \langle b | M | a \rangle
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\frac{dF}{d \langle a |} = \alpha M^T | b \rangle
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\frac{dF}{d | a \rangle} = \alpha \langle b | M
%}
{% F = \alpha \langle a | M | a \rangle
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\frac{dF}{d \langle a |} = \alpha (M + M^T) | a \rangle
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\frac{dF}{d | a \rangle} = \alpha \langle a |( M + M^T)
%}
see Mazzoni Table 3.1