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# The Hessian matrix

The Hessian matrix is a way of organizing all the second partial derivative information of a multivariable function. Created by Grant Sanderson.

## Want to join the conversation?

• How do you write the Hessian matrix notation by hand? Surely Boldface H is only used in printed form?

I mean, that's the case with vectors. Written by hand, you draw an arrow over the letter.

And does the Hessian matrix have anything to do with the symbol "Ĥ"? They look similar. (I found this symbol in Schrödinger's Equations, as in Ĥ|ɸ> = i ∂/∂t |ɸ>.) • There are numerous ways to denote the Hessian, but the most common form (when writing) is just to use a capital 'H' followed by the function (say, 'f') for which the second partial derivatives are being taken. For example, H(f).
It is not necessary to bold, but it does help.
The fact that it is capitalised helps in identifying the fact that it is a matrix.

Furthermore, the 'Ĥ' in Schrödinger's Equation in Quantum Mechanics is known as the Hamiltonian, which is different from the Hessian.

Hope that clears things up.
• I thought that constants become 0 when taking the derivate, the same way for instance a "4" would go away when taking a derivative. Why is that not the case here? • If you had f(x) = x + 4, then f'(x) = d/dx[x + 4] = d/dx[x] + d/dx = 1 + 0 = 1 - - - - the 4 "went away"
But
if you have f(x) = 4x, then f'(x) = d/dx[4x] = 4 d/dx[x] = (4)(1) = 4 - - - - here the 4 stays, why?

So if I had f(x,y) = e^x + sin(y), and wanted ∂f/∂x[f(x,y)], we would have
∂f/∂x[f(x,y)] = ∂f/∂x[e^x] + ∂f/∂x[sin(y)] = e^x + 0 = e^x, and the sin(y) "goes away"
But,
If you have f(x,y) = (e^x)(sin(y)) and want ∂f/∂x[f(x,y)], we can think of the sin(y) as a constant and remove it from the differential operator just like we did the 4 above . . . .
∂f/∂x[f(x,y)] = ∂f/∂x[(e^x)(sin(y)] = sin(y)∂f/∂x[e^x} = sin(y)e^x or (e^x)(sin(y)).

When dealing with variables that are being multiplied by a constant, we can take the constant out of the differential operator. In this case, sin(y) is a constant because in this example we are differentiating with respect to x

I hope that helped.
Stefen 