# Find a matrix such that the quadratic form of a orthonormal basis is equal to the Kronecker delta

For $$n \in \mathbb{N}$$ let $$\{v_1, v_2, \ldots v_n \}$$ be a orthonormal basis of $$\mathbb{R}^{n}$$. Further, let $$i \in \{ 1, 2, \ldots, n \}$$ be arbitrary but fixed. I am trying to prove that there exists a symmetric matrix $$M \in \mathbb{R}^{n \times n}$$ such that for all $$j \in \{ 1,2, \ldots, n \}$$ it holds that $$v_{j}^{T}Mv_{j} = \delta_{ij},$$ where $$\delta_{ij}$$ is the Kronecker delta.

Intuitively, I would assume such a Matrix $$M$$ exists, because we can arbitrarily choose all diagonal elements of $$M$$ and all the entries above the main diagonal. Thus we have $$\frac{n(n+1)}{2}$$ degrees of freedom and $$n$$ only equations that must hold. I tried to diagonalize $$M$$ to prove that the system has a solution. However, I have not been able to do so. Are there any solutions to the system or do I need more additional conditions on $$M$$?

• Does $M=I$ not work? The basis being orthonormal and all... maybe I'm very confused.
– Tom
Nov 28, 2023 at 23:42
• @Tom: $i$ is fixed. Nov 28, 2023 at 23:47
• Of course, sorry. Then I agree with @MartinArgerami
– Tom
Nov 29, 2023 at 0:19

Consider first the canonical basis $$\{e_1,\ldots,e_n\}$$. Then you can consider the matrix $$M=e_ie_i^T$$, that is the matrix with $$1$$ in the $$i,i$$ entry and zeroes everywhere else. Then $$e_j^TMe_j=\delta_{ij}$$.
Now let $$U$$ be the linear map induced by $$Uv_j=e_j$$, $$j=1,\ldots,n$$. Then $$U$$ is the unitary change of basis from the canonical basis to your basis. By the first paragraph there exists a matrix a symmetric matrix $$N$$ with $$e_j^TNe_j=\delta_{ij}$$. Put $$M=U^TNU$$. Then $$M$$ is symmetric and $$v_j^TMv_j=v_j^TU^TNUv_j=(Uv_j)^TNUv_j=e_j^*Ne_j=\delta_{ij}.$$
Note that the non-diagonal entries play no role in the first paragraph. In other words, one can choose any $$M$$ with zero diagonal, other than the $$1$$ in the $$i^{\rm th}$$ entry, and all the non-diagonal entries are only constrained by the symmetry condition.