Matrices form an inner-product space

Dependencies:

1. Matrices over a field form a vector space
2. Inner product space
3. Dot-product of vectors
4. Trace of a matrix
5. /complex-numbers/conjugation-is-homomorphic
6. /complex-numbers/conjugate-product-abs
7. Transpose of product

Let $F$ be a subfield of $\mathbb{C}$ (complex numbers). Then ${\mathbb{M}}_{m,n}(F)$ is an inner product space.

The inner product of matrices is given by $\mathrm{tr}(B^{\ast }A)$, where $A^{\ast }$ is the conjugate transpose of $A$.

$$A^{\ast }[i,j]=\overline{A[j,i]}⟹A^{\ast }={\bar{A}}^T=\overline{A^T}$$

If we only consider column vectors ($n=1$), $$⟨\mathbf{u},\mathbf{v}⟩=\mathrm{tr}({\mathbf{v}}^{\ast }\mathbf{u})={\mathbf{v}}^{\ast }\mathbf{u}=\mathbf{v}\cdot \mathbf{u}$$ which is the dot product of $\mathbf{v}$ and $\mathbf{u}$.

For real-valued matrices, $⟨A,B⟩=\mathrm{tr}(A^{T}B)$ is an equivalent definition of inner product.

Proof

Matrices over a field form a vector space. We only need to prove properties of the inner-product.

• Conjugate symmetry: $$\begin{array}{rl}⟨A,B⟩& =\mathrm{tr}(B^{\ast }A)\\ & =\sum _{i=1}^n(B^{\ast }A)[i,i]\\ & =\sum _{i=1}^n\sum _{j=1}^m{B}^{\ast }[i,j]A[j,i]\\ & =\sum _{i=1}^n\sum _{j=1}^m\overline{B[j,i]}A[j,i]\\ \text{(}\overline{\stackrel{―}{z}}=z\mathrm{\forall }z\in \mathbb{C}\text{)}& & =\sum _{i=1}^n\sum _{j=1}^m\overline{B[j,i]}\overline{\stackrel{―}{A[j,i]}}\text{(conjugation is homomorphic)}& & =\sum _{i=1}^n\sum _{j=1}^m\overline{B[j,i]\stackrel{―}{A[j,i]}}& =\sum _{i=1}^n\sum _{j=1}^m\overline{\stackrel{―}{A[j,i]}B[j,i]}\\ & =\sum _{i=1}^n\sum _{j=1}^m\overline{A^{\ast }[i,j]B[j,i]}\\ & =\sum _{i=1}^n\overline{(A^{\ast }B)[i,i]}\\ \text{(conjugation is homomorphic)}& & =\overline{\sum _{i=1}^n(A^{\ast }B)[i,i]}& =\overline{\mathrm{tr}(A^{\ast }B)}=\overline{⟨B,A⟩}\end{array}$$

• Linearity in first argument: $$\begin{array}{rl}⟨A+B,C⟩& =\mathrm{tr}(C^{\ast }(A+B))\\ \text{(distributivity)}& & =\mathrm{tr}(C^{\ast }A+C^{\ast }B)& =\mathrm{tr}(C^{\ast }A)+\mathrm{tr}(C^{\ast }B)\\ & =⟨A,C⟩+⟨B,C⟩\end{array}$$ $$\begin{array}{rl}⟨cA,B⟩& =\mathrm{tr}(B^{\ast }(cA))\\ \text{(associativity)}& & =\mathrm{tr}(c(B^{\ast }A))& =c\mathrm{tr}(B^{\ast }A)\\ & =c⟨A,B⟩\end{array}$$

• Positive-definiteness: $$\begin{array}{rl}⟨A,A⟩& =\mathrm{tr}(A^{\ast }A)\\ & =\sum _{i=1}^n(A^{\ast }A)[i,i]\\ & =\sum _{i=1}^n\sum _{j=1}^m{A}^{\ast }[i,j]A[j,i]\\ & =\sum _{i=1}^n\sum _{j=1}^m\overline{A[j,i]}A[j,i]\\ & =\sum _{i=1}^n\sum _{j=1}^m|A[j,i]{|}^2\end{array}$$ Therefore, $⟨A,A⟩\ge 0$. $$\begin{array}{rl}⟨A,A⟩=0& ⟺\sum _{i=1}^n\sum _{j=1}^m|A[j,i]{|}^2=0⟺\mathrm{\forall }i,\mathrm{\forall }j,|A[j,i]{|}^2=0\\ & ⟺\mathrm{\forall }i,\mathrm{\forall }j,A[j,i]=0⟺A=0\end{array}$$

Therefore, ${\mathbb{M}}_{m,n}(F)$ is an inner-product space.

If we only consider real-valued matrices, $$⟨A,B⟩=\mathrm{tr}(B^{\ast }A)=\mathrm{tr}(B^{T}A)=\mathrm{tr}((A^{T}B{)}^T)=\mathrm{tr}(A^{T}B)$$

Dependency for:

1. Matrix of orthonormal basis change
2. Symmetric operator iff hermitian
3. Orthogonal matrix

Info:

• Depth: 5
• Number of transitive dependencies: 14

Transitive dependencies:

1. /complex-numbers/conjugate-product-abs
2. /complex-numbers/conjugation-is-homomorphic
3. Group
4. Ring
5. Vector
6. Dot-product of vectors
7. Field
8. Vector Space
9. Inner product space
10. Semiring
11. Matrix
12. Transpose of product
13. Trace of a matrix
14. Matrices over a field form a vector space