inner product

inner-product.md

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+# Inner Product
+
+- operates on two vectors and produces a scalar
+- notation $`\braket{x|y}`$
+  - $`\bra{x}`$ is a [[bra]]
+  - $`\ket{y}`$ is a [[ket]]
+- follows laws
+  1. $`\braket{0|y} = 0`$ and $`\braket{x|0} = 0`$
+  2. $`\braket{x + y|z} = \braket{x|z} + \braket{y|z}`$ and $`\braket{x|y + z} = \braket{x|y} + \braket{x|z}`$
+  3. $`\braket{cx|y} = \overline{c}\braket{x|y}`$ and $`\braket{x|cy} = \braket{x|y}c`$
+  4. $`\braket{x|y} = \overline{\braket{y|x}}`$
+- the inner product $`\braket{x|y}`$ is antilinear in $`x`$ and linear in $`y`$
+  - if starting with two [[ket]]s, take the conjugate transpose of the former: $`\bra{x} = \overline{\ket{x}}^\intercal = \overline{\ket{x}^\intercal}`$