Log in Start free

Part BCSIR NET June 2023positive-definiteness

Positive definiteness

Let $x = (x_{1}, \dots, x_{n})$ and $y = (y_{1}, \dots, y_{n})$ denote vectors in $R^{n}$ for a fixed n ≥ 2. Which of the following defines an inner product on $R^{n}$ ?

A.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $x_{i} y_{j}$
B.⟨x, y⟩ $= \sum_{i, j = 1}^{n}$ $(x_{i}^{2} + y_{j}^{2})$
C.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $j^{3} x_{j} y_{j}$ ✓
D.⟨x, y⟩ $= \sum_{j = 1}^{n}$ $x_{j} y_{n}$ ₋ $_{j}$ ₊ $_{1}$

Solution

A weighted sum $\sum w_{j} x_{j} y_{j}$ with positive weights is an inner product. (1) is $(\sum x_{i}) (\sum y_{j})$ , not positive definite; (2) is not bilinear; (4) is symmetric but ⟨x, x⟩ can be ≤ 0 (e.g. x = (1, −1) for n = 2).

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: A real matrix with all real eigenvalues is orthogonally diagonalisable

From Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

ShareWhatsApp Telegram