Inner Product

Definition: Vectors

Let $\mathbf{u},\mathbf{v}$ : Vectors in vector space $R^n$

Then the inner product of $\mathbf{u}$ and $\mathbf{v}$ is

$$⟨\mathbf{u},\mathbf{v}⟩=\mathbf{u}\cdot \mathbf{v}=u_1{v}_1+u_2{v}_2+\dots ,u_n{v}_n$$

Definition: Matrices

Let

• $\mathcal{V}$ : Linear space of $m\times n$ matrices
• $A,B$ : $m\times n$ matrices in $\mathcal{V}$

Then the inner product of $A$ and $B$ is

$$A\cdot B=\mathrm{tr}(A^{T}B)=\mathrm{tr}(A{B}^T)$$