Matrix Representations of inner products

Derivation of the Matrix Representation

Expressing the Inner Product in Terms of Coordinates

• For vectors $x,y\in V$, expressed as:

$$x=\sum _{i=1}^n{x}_i{v}_i\text{and}y=\sum _{j=1}^n{y}_j{v}_j,$$

• where $x_i$ and $y_j$ are coordinates of $x$ and $y$ with respect to basis $\alpha =\{v_1,\dots ,v_n\}$.

• Inner product $⟨x,y⟩$ is:

  $$⟨x,y⟩=⟨\sum _{i=1}^n{x}_i{v}_i,\sum _{j=1}^n{y}_j{v}_j⟩.$$

• By linearity of inner products:

  $$⟨x,y⟩=\sum _{i=1}^n\sum _{j=1}^n{x}_i{y}_j⟨v_i,v_j⟩.$$

Defining the Matrix $A$

• Define $a_{ij}=⟨v_i,v_j⟩$ for $i,j=1,\dots ,n$.
• Matrix $A=[a_{ij}]$ is symmetric because $⟨v_i,v_j⟩=⟨v_j,v_i⟩$.

Rewriting the Inner Product in Matrix Form

• Substitute $a_{ij}=⟨v_i,v_j⟩$ into the double sum:

• $$⟨x,y⟩=\sum _{i=1}^n\sum _{j=1}^n{x}_i{y}_j{a}_{ij}.$$

• Matrix Notation:

  • Let $[x{]}_{\alpha }$ and $[y{]}_{\alpha }$ be coordinate vectors of $x$ and $y$ with respect to basis $\alpha$: $$[x{]}_{\alpha }=\left[\begin{array}{c}{x}_1\\ x_2\\ ⋮\\ x_n\end{array}\right],[y{]}_{\alpha }=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right].$$
  • The inner product can now be written as: $$⟨x,y⟩=[x{]}_{\alpha }^{T}A[y{]}_{\alpha }.$$

Interpreting the Matrix Representation

• Matrix $A$ is called the matrix representation of the inner product with respect to the basis $\alpha$.
• The expression $[x{]}_{\alpha }^{T}A[y{]}_{\alpha }$ allows the inner product $⟨x,y⟩$ to be computed using:
  • Coordinates of $x$ and $y$ with respect to $\alpha$.
  • The symmetric matrix $A$ that encodes the inner products $⟨v_i,v_j⟩$.

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For any symmetric matrix $A$ and for a fixed basis $\alpha$, the formula $⟨x,y⟩=[x{]}_{\alpha }^{T}A[y{]}_{\alpha }$ seems to give rise to an inner product on $V$. In fact, the formula clearly satisfies the first three rules in the definition of the inner product, but not necessarily the fourth rule, positive definiteness. The following theorem gives a necessary condition for a symmetric matrix $A$ to give rise to an inner product.

Another Condition for $A$ for it to be a Matrix Representation of a vector space $V$

Statement: The matrix representation $A$ of an inner product with respect to any basis on a vector space $V$ is invertible.

Proof Strategy

The proof shows the column vectors of $A$ are linearly independent, which is sufficient for invertibility.

Detailed Proof Steps

1. Setup:

   • Let $\alpha =\{v_1,...,v_n\}$ be a basis for inner product space $V$
   • Matrix $A=[a_{ij}]=[⟨v_i,v_j⟩]$
   • Column vectors denoted as $a_j$ for $j=1,...,n$

2. Linear Dependence Check:

   • Consider a linear combination of columns equaling zero: $0=c_1{a}_1+...+c_n{a}_n$ where $c_i\in \mathbb{R}$

3. Key Transform:

   • Define vector $c={\sum }_{i=1}^n{c}_i{v}_i\in V$
   • In basis coordinates: $[c{]}_{\alpha }=[c_1\cdots c_n{]}^T$

4. System Analysis:

   • The equation becomes a homogeneous system:

$$\{\begin{array}{l}0=a_{11}{c}_1+...+a_{1n}{c}_n={\sum }_{j=1}^n{a}_{1j}{c}_j=[v_1{]}_{\alpha }^{T}A[c{]}_{\alpha }=⟨v_1,c⟩\\ ⋮\\ 0=a_{n1}{c}_1+...+a_{nn}{c}_n={\sum }_{j=1}^n{a}_{nj}{c}_j=[v_n{]}_{\alpha }^{T}A[c{]}_{\alpha }=⟨v_n,c⟩\end{array}$$

5. Critical Step:

   • Using $[v_i{]}_{\alpha }=e_i$ (standard basis vectors)
   • By Corollary 5.3 (referenced in proof), we get: $c={\sum }_{i=1}^n{c}_i{v}_i=0$

6. Conclusion:

   • This shows $c_1=c_2=...=c_n=0$
   • Therefore, the columns of $A$ are linearly independent
   • Thus, $A$ is invertible