In real coordinates the standard inner product is $\langle x,y\rangle=x^Ty$. In complex coordinates use $\langle x,y\rangle=x^*y=\sum_j\overline{x_j}y_j$. This graph uses the convention conjugate-linear in the first argument, linear in the second. Inner products define norms, orthogonality, angles in real spaces, and projections. Plain $x^Ty$ is not positive definite on $\mathbb C^n$.
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- Definition of inner/dot product on R^nFor $x,y\in\mathbb R^n$, the dot product is $\langle x,y\rangle=\sum jx jy j=x^Ty$. It is symmetric, bilinear and...
- Definition of inner/dot product on C^nFor $x,y\in\mathbb C^n$, use $\langle x,y\rangle=\sum j\overline{x j}y j=x^ y$. It is conjugate-linear in $x$, linear...
- The standard inner product on R^n is commutative.For real vectors, $\langle x,y\rangle=\langle y,x\rangle$, because scalar products $x jy j$ commute. This symmetry is...
- The complex inner product is conjugate symmetricThe complex inner product is conjugate symmetric , not anticommutative: $\langle x,y\rangle=\overline{\langle...
- The standard inner product on R^n commutes with (real) scalar multiplication.For $a\in\mathbb R$, $\langle ax,y\rangle=a\langle x,y\rangle$ and $\langle x,ay\rangle=a\langle x,y\rangle$....
- Complex inner products conjugate scalars in the first argumentWith $\langle x,y\rangle=x^ y$ and $a\in\mathbb C$, $\langle ax,y\rangle=\overline a\langle x,y\rangle$, while...
- The standard inner product on R^n (or C^n) distributes over addition.Inner products are additive in both slots: $\langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle$ and $\langle...
- The standard inner product of a vector with itself is non-negativeFor a standard real or complex inner product, $\langle x,x\rangle=\sum j x j ^2\ge0$. The value is always real, even...
- The standard inner product of a vector with itself is 0 only for the 0 vectorThe sum $\langle x,x\rangle=\sum j x j ^2$ is zero iff every coordinate is zero. This is positive definiteness....
- The standard inner product on R^n can be written as the product of a vector and the transpose of a vector.For real column vectors, the dot product equals $x^Ty$, a $1\times1$ matrix identified with a scalar. By contrast,...
- The standard inner product on C^n can be written as the product of a vector and the adjoint of a vector.For complex column vectors, $\langle x,y\rangle=x^ y$. The conjugate transpose is necessary for positivity. The outer...
- A matrix turns into its adjoint when moved to the other side of the standard inner product on C^n.For $A\in\mathbb C^{m\times n}$, $\langle Ax,y\rangle=\langle x,A^ y\rangle$. Indeed, $(Ax)^ y=x^ A^ y$. Likewise...
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