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 $\operatorname{tr}(B^*A)$, where $A^*$ is the conjugate transpose of $A$.

\[ A^*[i, j] = \overline{A[j, i]} \implies A^* = \overline{A}^T = \overline{A^T} \]

If we only consider column vectors ($n=1$), \[ \langle \mathbf{u}, \mathbf{v} \rangle = \operatorname{tr}(\mathbf{v}^*\mathbf{u}) = \mathbf{v}^*\mathbf{u} = \mathbf{v} \cdot \mathbf{u} \] which is the dot product of $\mathbf{v}$ and $\mathbf{u}$.

For real-valued matrices, $\langle A, B \rangle = \operatorname{tr}(A^TB)$ 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{align} \langle A, B \rangle &= \operatorname{tr}(B^*A) \\ &= \sum_{i=1}^n (B^*A)[i, i] \\ &= \sum_{i=1}^n \sum_{j=1}^m B^*[i, j]A[j, i] \\ &= \sum_{i=1}^n \sum_{j=1}^m \overline{B[j, i]}A[j, i] \\ &= \sum_{i=1}^n \sum_{j=1}^m \overline{B[j, i]}\;\overline{\overline{A[j, i]}} \tag{$\overline{\overline{z}} = z \forall z \in \mathbb{C}$} \\ &= \sum_{i=1}^n \sum_{j=1}^m \overline{B[j, i]\overline{A[j, i]}} \tag{conjugation is homomorphic} \\ &= \sum_{i=1}^n \sum_{j=1}^m \overline{\overline{A[j, i]}B[j, i]} \\ &= \sum_{i=1}^n \sum_{j=1}^m \overline{A^*[i, j]B[j, i]} \\ &= \sum_{i=1}^n \overline{(A^*B)[i, i]} \\ &= \overline{\sum_{i=1}^n (A^*B)[i, i]} \tag{conjugation is homomorphic} \\ &= \overline{\operatorname{tr}(A^*B)} = \overline{\langle B, A \rangle} \end{align}

• Linearity in first argument: \begin{align} \langle A+B, C \rangle &= \operatorname{tr}(C^*(A+B)) \\ &= \operatorname{tr}(C^*A + C^*B) \tag{distributivity} \\ &= \operatorname{tr}(C^*A) + \operatorname{tr}(C^*B) \\ &= \langle A, C \rangle + \langle B, C \rangle \end{align} \begin{align} \langle cA, B \rangle &= \operatorname{tr}(B^*(cA)) \\ &= \operatorname{tr}(c(B^*A)) \tag{associativity} \\ &= c\operatorname{tr}(B^*A) \\ &= c\langle A, B \rangle \end{align}

• Positive-definiteness: \begin{align} \langle A, A \rangle &= \operatorname{tr}(A^*A) \\ &= \sum_{i=1}^n (A^*A)[i, i] \\ &= \sum_{i=1}^n \sum_{j=1}^m A^*[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{align} Therefore, $\langle A, A \rangle \ge 0$. \begin{align} \langle A, A \rangle = 0 &\iff \sum_{i=1}^n \sum_{j=1}^m |A[j, i]|^2 = 0 \iff \forall i, \forall j, |A[j, i]|^2 = 0 \\ &\iff \forall i, \forall j, A[j, i] = 0 \iff A = 0 \end{align}

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

If we only consider real-valued matrices, \[ \langle A, B \rangle = \operatorname{tr}(B^*A) = \operatorname{tr}(B^TA) = \operatorname{tr}((A^TB)^T) = \operatorname{tr}(A^TB) \]

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