Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 5

Table of contents
- a 5
  - Solution a
- b 5
  - Solution b

a

Solution

The Kronecker delta restricts the Einstein summation convention to equal indices, giving the dot product

δ_{ij} a_{i} b_{j} = \sum_{i} a_{i} b_{i} = a \cdot b .

b

Solution

By the definitions of matrix multiplication, transpose and trace,

tr (C^{T} D) = \sum_{i} \sum_{j} (C^{T})_{ij} D_{ji} = \sum_{i} \sum_{j} C_{ji} D_{ji} .

Relabelling the summed indices gives

C_{ij} D_{ij} = tr (C^{T} D) = tr (C D^{T}),

the Frobenius inner product of these real matrices.

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