🔬 Tutorial problems gamma \gamma

🔬 Tutorial problems gamma \(\gamma\)#

Note

This problems are designed to help you practice the concepts covered in the lectures. Not all problems may be covered in the tutorial, those left out are for additional practice on your own.

\(\gamma\).1#

Consider an \((n \times n)\) Vandermonde matrix [this one can be named :)] of the form

\[\begin{split} V = \begin{bmatrix} 1 & 1 & \cdots & 1 \\ x_1 & x_2 & \cdots & x_n \\ x_1^2 & x_2^2 & \cdots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_1^{n-1} & x_2^{n-1} & \cdots & x_n^{n-1} \end{bmatrix} \end{split}\]

Show that the determinant of \(V\) is given by

\[ \det(V) = \Pi_{j<i \leqslant n}(x_i-x_j) \]

for the cases \(n=2\), \(n=3\) and \(n=4\)

Properties of the determinants help in finding an elegant solution.

\(\gamma\).2#

Consider a function \(f : \mathbb{R}^N \ni x \mapsto x^{T}Bx \in \mathbb{R}\), where \(N \times N\) matrix \(B\) is square but not symmetric.

Show that the same function can be represented as \(x^{T}Ax\) where \(A\) is symmetric.

Given a square matrix \(M\), you can use the identity \(M = \tfrac{1}{2}(M+M') + \tfrac{1}{2}(M-M')\) where the first component is symmetric and the second is not symmetric.

If \(A\) and \(B\) are conformable for matrix multiplication, then \((AB)^{T} = B^T A^T\).

\(\gamma\).3#

In which direction should one move from a given point in order to increase the value of the function most rapidly:

  1. \(\quad\) \(f(x,y) = 4x^2y\) from the point \((2,3)\)

  2. \(\quad\) \(f(x,y) = y^2 e^{3x}\) from the point \((0,3)\)

Present your answer as a vector of length 1.

[Simon and Blume, 1994]: Exercises 14.18, 14.19

Review the definition and facts about the gradient of a multivariate functions.

\(\gamma\).4#

A critical point of a multivariate function is the point at which all partial derivatives are zero.

Compute the critical points of the following functions:

  1. \(\quad\) \(x^4+x^2-6xy + 3y^2\)

  2. \(\quad\) \(x^2-6xy+2y^2+10x+2y-5\)

  3. \(\quad\) \(xy^2+x^3y-xy\)

  4. \(\quad\) \(3x^4+3x^2y-y^3\)

  5. \(\quad\) \(x^2+6xy+y^2-3yz+4z^2-10x-5y-21z\)

  6. \(\quad\) \((x^2+2y^2+3z^2) e^{-(x^2+y^2+z^2)}\)

[Simon and Blume, 1994]: Exercises 17.1, 17.2

\(\gamma\).5#

Consider a function \(f : \mathbb{R}^N \ni x \mapsto x'Ax \in \mathbb{R}\), where \(N \times N\) matrix \(A\) is symmetric.

Using the product rule of multivariate calculus, derive the gradient and Hessian of \(f\). Make sure that all multiplied vectors and matrices are conformable.

You can assume that \(x\) is a column vector, and that any vector function of \(x\) is also a column vector.

\(\gamma\).6#

Compute directional derivative of \(f(x,y) = xy^2 + x^3y\) at the point \((4,-2)\) in the direction of the vector \((1/\sqrt{10},3/\sqrt{10})\).

Proceed in two different ways:

  • first, using the definition of the directional derivative, write down a function \(g \colon \mathbb{R} \to \mathbb{R}\) of \(h\) given as a slice of the original function \(f(x,y)\) through the given point in the given direction; then differentiate this function and compute the derivative at \(h=0\)

  • second, use the gradient formula; and verify that the same answer is obtained

[Simon and Blume, 1994]: Exercise 14.20

Follow the example in the lecture notes.

Contents