- #1
perplexabot
Gold Member
- 329
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Hey all. Let me just get right to it! Assume you have a function [itex]f:\mathbb{R}^n\rightarrow\mathbb{R}^m[/itex] and we know nothing else except the following equation:
[itex]\triangledown_x\triangledown_x^Tf(x)^TQy=0[/itex]
where [itex]\triangledown_x[/itex] is the gradient with respect to vector [itex]x[/itex] (outer product of two gradient operators is the hessian operator). Also let the dimensions of [itex]Q[/itex] and [itex]y[/itex] conform.
Using the information provided above what can you conclude about [itex]f(x)[/itex] (if anything)? Can you infer that [itex]f(x)[/itex] is linear?
Thank you : )
[itex]\triangledown_x\triangledown_x^Tf(x)^TQy=0[/itex]
where [itex]\triangledown_x[/itex] is the gradient with respect to vector [itex]x[/itex] (outer product of two gradient operators is the hessian operator). Also let the dimensions of [itex]Q[/itex] and [itex]y[/itex] conform.
Using the information provided above what can you conclude about [itex]f(x)[/itex] (if anything)? Can you infer that [itex]f(x)[/itex] is linear?
Thank you : )