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Linear Algebra : The Hessian

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Example Question #41 : Linear Algebra

Let $f(x,y, z) = xy \cos z + xz \tan y$

Which of the following does not appear in the Hessian matrix of ?

Possible Answers:

$- x \sin z + x \sec^{2} y$

$- xy \cos z$

$- y \sin z + \tan y$

$\cos z + z \sec^{2} y$

$2 xz \sec^{3} y$

Correct answer:

$2 xz \sec^{3} y$

Explanation:

The Hessian matrix of is the matrix of partial second derivatives

$H(f) = \begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$

To identify which choice does not give an entry of the matrix, we need to find all nine partial derivatives; however, since $f_{yx}= f_{xy}$ , $f_{zx}= f_{xz}$ , and $f_{zy}= f_{yz}$ , we need only find six such derivatives. They are as follows:

$f(x,y, z) = xy \cos z + xz \tan y$

$f_{x}(x,y, z) = y \cos z + z \tan y$

$f_{xx}(x,y, z) =0$

$f_{xy}(x,y, z) = \cos z + z \sec^{2} y$

$f_{xz}(x,y, z) =- y \sin z + \tan y$

$f_{y}(x,y, z) = x \cos z + xz \sec^{2} y$

$f_{yy}(x,y, z) = 2 xz \tan y \sec^{2} y$

$f_{yz}(x,y, z) = - x \sin z + x \sec^{2} y$

$f_z(x,y, z) =- xy \sin z + x \tan y$

$f_{zz}(x,y, z) =- xy \cos z$

Of the five given choices, only $2 xz \sec^{3} y$ is not one of the partial second derivatives. This is the correct choice.

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Example Question #32 : The Hessian

Let $f(x, y) =2^{x}3^{y}$ .

Find the Hessian matrix H(f) .

Possible Answers:

$2^{x}3^{y} \begin{bmatrix} (\ln 2)^{2} &1 \\1& (\ln 3)^{2} \end{bmatrix}$

$2^{x}3^{y} \begin{bmatrix} (\ln 2)^{2} &(\ln 2)(\ln 3) \\ (\ln 2)(\ln 3) & (\ln 3)^{2} \end{bmatrix}$

$2^{x}3^{y} \begin{bmatrix} 1 & 0 \\ 0& 1 \end{bmatrix}$

$2^{x}3^{y} \begin{bmatrix} 1 &(\ln 2)(\ln 3) \\ (\ln 2)(\ln 3) & 1 \end{bmatrix}$

$2^{x}3^{y} \begin{bmatrix} 1& 1 \\ 1 & 1 \end{bmatrix}$

Correct answer:

$2^{x}3^{y} \begin{bmatrix} (\ln 2)^{2} &(\ln 2)(\ln 3) \\ (\ln 2)(\ln 3) & (\ln 3)^{2} \end{bmatrix}$

Explanation:

The Hessian matrix is the matrix of partial second derivatives

$H(f)=\begin{bmatrix} f_{xx}(x,y) & f_{xy}(x,y) \\ f_{yx}(x,y) & f_{yy}(x,y) \end{bmatrix}$ .

$f(x, y) =2^{x}3^{y}$ can be rewritten as

$f(x, y) =(e^{ \ln 2} )^{x}\cdot (e^{ \ln 3})^{y}$

$f(x, y) =e^{x \ln 2} \cdot e^{y \ln 3}$ , then

$f(x, y) =e^{x \ln 2+ y \ln 3}$

This makes the partial second derivatives easier to find.

$f_{x}(x, y) =(\ln 2) e^{x \ln 2+ y \ln 3}$

$f_{xx}(x, y) =(\ln 2) (\ln 2)e^{x \ln 2+ y \ln 3} = 2^{x}3^{y}(\ln 2)^{2}$

$f_{xy}(x, y) =(\ln 2) (\ln 3)e^{x \ln 2+ y \ln 3}= 2^{x}3^{y}(\ln 2)(\ln 3)$

$f_{y}(x, y) =(\ln 3) e^{x \ln 2+ y \ln 3}$

$f_{yx}(x, y) =(\ln 3) (\ln 2)e^{x \ln 2+ y \ln 3}= 2^{x}3^{y}(\ln 2)(\ln 3)$

$f_{yy}(x, y) =(\ln 3) (\ln 3)e^{x \ln 2+ y \ln 3}= 2^{x}3^{y}(\ln 3)^{2}$

The Hessian matrix for is

$H(f)=\begin{bmatrix} 2^{x}3^{y}(\ln 2)^{2} & 2^{x}3^{y}(\ln 2)(\ln 3) \\ 2^{x}3^{y}(\ln 2)(\ln 3) & 2^{x}3^{y}(\ln 3)^{2} \end{bmatrix}$

$= 2^{x}3^{y} \begin{bmatrix} (\ln 2)^{2} &(\ln 2)(\ln 3) \\ (\ln 2)(\ln 3) & (\ln 3)^{2} \end{bmatrix}$

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Example Question #41 : Matrix Calculus

Consider the function $f(x, y, z ) = \cos x \sin y \tan z$ .

Which of the following expressions does not appear twice in the Hessian matrix of ?

Possible Answers:

$- \cos x \sin y \tan z$

$2 \cos x \sin y \tan z \sec^{2} z$

$\cos x \cos y \sec^{2} z$

$- \sin x \cos y \tan z$

$- \sin x \sin y \sec^{2} z$

Correct answer:

$2 \cos x \sin y \tan z \sec^{2} z$

Explanation:

The Hessian matrix is the matrix of partial second derivatives

$H(f)=\begin{bmatrix} f_{xx} & f_{xy} & f_{xz} \\ f_{yx} & f_{yy} & f_{yz} \\ f_{zx} & f_{zy} & f_{zz} \end{bmatrix}$ .

Since $f_{yx}= f_{xy} , f_{zx}= f_{xz} , f_{zy}= f_{yz}$ , we only need to find six partial second derivatives and compare them to the five choices.

$f = \cos x \sin y \tan z$

$f_{x} =- \sin x \sin y \tan z$

$f_{xx} =- \cos x \sin y \tan z$

$f_{xy} =- \sin x \cos y \tan z$

$f_{xz} =- \sin x \sin y \sec^{2} z$

$f_{y} = \cos x \cos y \tan z$

$f_{yy} = - \cos x \sin y \tan z$

$f_{yz} = \cos x \cos y \sec^{2} z$

$f_{z} = \cos x \sin y \sec^{2} z$

$f_{zz} =2 \cos x \sin y \tan z \sec^{2} z$

As stated before,

$f_{yx}= f_{xy} = - \sin x \cos y \tan z$

$f_{zx}= f_{xz} =- \sin x \sin y \sec^{2} z$

$f_{zy}= f_{yz} = \cos x \cos y \sec^{2} z$ ,

so all three expressions will appear twice in the Hessian matrix.

Also note that

$f_{xx} = f_{yy} =- \cos x \sin y \tan z$ ,

so this expression appears twice as well.

$f_{zz} =2 \cos x \sin y \tan z \sec^{2} z$ ,

however, only appears once. This is the correct choice.

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Linear Algebra : The Hessian

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #41 : Linear Algebra

Example Question #32 : The Hessian

Example Question #41 : Matrix Calculus

All Linear Algebra Resources

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