A matrix called the Vandermonde matrix, appears in a number of places from statistics, polynomial interpolation (and extensively in error control coding, Fast Fourier transforms, and signal processing). It is also a very elegant matrix from a mathematical point of view. A \(3\times 3\) Vandermonde matrix is
\[\begin{bmatrix} 1 & x_1 & x_1^2 \\ 1 & x_2 & x_2^2 \\ 1& x_3 & x_3^2 \end{bmatrix}\]Find the determinant of the matrix above.
Show that the determinant of
\[\begin{bmatrix} a & b & c \\ b & c & a \\ c& a & b \end{bmatrix}\]equals 0 iff \(a=b=c\) or if \(a+b+c = 0\).
Let
\[P_1 = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0& 1 \end{bmatrix} \textrm{ and } P_2 = \begin{bmatrix} 0 & 0& 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{bmatrix}\]Find all values of \(\lambda\) such that \(P_1-\lambda I_3\) and all \(\mu\) such that \(P_2-\mu I_3\) are singular.
Jacobian Let \(x = f_1(u, v)\) and \(y = f_2(u,v)\). We think of \((u,v)\) to be the coordinates in a 2-\(d\) plane, and \(f_1\) and \(f_2\) are (possibly non-linear) transformations of the coordinates. Assume that the transformations map each point \((u,v)\) to a unique point \((x,y)\) to visualize this better.If \((u,v)\) maps to \((x,y)\), recall from elementary calculus that an elemental change \((u+du, v+dv)\) changes the mapped point to \((x+\Delta x, y+\Delta y)\), where: \(\begin{align*} \Delta x = \frac{\partial f_1}{\partial u } du + \frac{\partial f_1}{\partial v} dv\\ \Delta y = \frac{\partial f_2}{\partial u } du + \frac{\partial f_2}{\partial v} dv.\\\)\end{align*}
Imagine the elemental rectangle with corners \((u,v)\), \((u+ du, v)\), \((u, v+dv)\) and \((u+du, v+dv)\). Find the point in the \(x,y\) plane that each corner of the rectangle above maps to. Since we are working with elemental increments, the rectangle in the \(u,v\) plane maps in general to a parallelogram in the \(x,y\) plane. Call this parallelogram as \(D_{x,y}\)
What is the ratio of area of the elemental parallelogram \(D_{x,y}\) to \(du dv\), the area of the elemental rectangle in the \((u,v)\)plane?
This ratio of areas is called the Jacobian, and you will encounter it again in EE342, among other topics.
Cramer’s Rule Cramer’s rule is a neat way to combine what we learnt about solving equations with what we know about determinants, at least in some cases. Suppose \(A\) is a square \(n\times n\) matrix, \({\bf b}\) is a \(n\times 1\) vector, and we are trying to solve \(A{\bf x} ={\bf b}\). If \(A\) is rectangular, Cramer’s rule does not help us find solutions if any. For the following, you may choose to do the problems for \(3\times 3\) matrix \(A\) and a \(3\times 1\) vector \(\bf b\), and convince yourself that the arguments generalize for any \(n\times n\) matrix.
If \(A\) is singular, what is the determinant of \(A\)? In this case, \(A{\bf x}={\bf b}\) has potentially infinite solutions or no solution, depending on whether \({\bf b}\) is a linear combination of the columns of \(A\) or not. We are out of luck here, Cramer’s rule does not help us find a solution even if there is one.
If \(A\) is not singular, we know the determinant of \(A\) is not 0. What can you say about the solutions of \(A{\bf x} = {\bf b}\)? Why?
Suppose \(A\) is not singular, and let \(A{\bf x}_1 ={\bf b}\), where \({\bf x}_1\) is the unique solution of \(A{\bf x}={\bf b}\). Let \(X_1\) be a \(n\times n\) matrix, whose first column is \({\bf x}_1\), and the remaining \(n\) columns are the corresponding columns of \(I_n\). What is \(AX_1\)?
Cramer’s rule: Show that the determinant of \(X_1\) is simply the first component of \({\bf x}_1\).
Extend the above argument to obtain all other components of \({\bf x}_1\).
Extend the above argument to find the inverse of a square, non-singular matrix \(A\).