Homework 2

Due Oct 11th at 11:59pm. Submit to CANVAS at https://canvas.uw.edu/courses/1828831/assignments/10766564.

Problem 1

Let \(A\) and \(B\) be real \(n\) by \(n\) matrices. Suppose both are nonsingular.

a) Show that \(AB\) is nonsingular.

b) Find an example where \(A+B\) is singular.

Problem 2

We say a real symmetric matrix is positive semidefinite if \(x^T A x \geq 0\) for all \(x\).

a) Find a matrix that is positive semidefinite but not positive definite.

b) Show that for an arbitrary matrix \(B\), the matrix \(B^T B\) is positive semidefinite.

Problem 3

An extremely useful fact about symmetric matrices is that their eigenvectors always form an orthogonal basis. That is, if a \(n\) by \(n\) real matrix \(A\) is symmetric, it has \(n\) eigenvectors \(u_1, \dots, u_n\) and \(u_i^T u_j =0\) if \(i \neq j\). This holds for complex matrices if we replace symmetric by hermitian. We won’t prove this statement here. It can also be generalized significantly to include infinite dimensional operators. For the following problem, we look at how it can be used.

a) Show that all eigenvalues of a positive definite matrix are positive and all eigenvalues of a positive semidefinite matrix are nonnegative.

b) Let \(A\) be a symmetric matrix. What is the largest \(\lambda\) such that \(A - \lambda I \succ 0\). Recall that \(\succ\) means positive definiteness and \(I\) is the identity matrix.

Problem 4

The notation of positive definiteness is usually defined for symmetric matrices. We can extend to nonsymmetric square matrices using the same definition as before: given a \(n\) by \(n\) matrix \(A\) (not necessarily symmetric), we say \(A\) is positive definte if \(x^T A x >0\) for all nonzero vectors \(x\).

a) Show that \(A\) is positive definite if and only if \(A+A^T\) is positive definite.

b) Suppose \(A\) and \(B\) and symmetric postive definite matrices, is \(AB\) positive semidefinite?

Problem 5

Consider the following network.

a) Suppose the voltages are \(V_1=1+0.1j, V_2=0.9+0.5j, V_3=-0.8+0.2j, V_4=-0.3+j, V_5=1-0.2j\). Find the current injections.

b) Suppose the current injections are \(I_1=2, I_2=-j, I_3=0.5-2j, I_4=0.5+0.5j, I_5=1-j\). Find the bus voltages. Hint: check whether the \(Y\)-bus matrix is invertable.

Problem 6

Schur complement can be used to reduce the size of circuits by removing internal buses. This is called Kron reduction. We will work through it for the following circuit, with the goal of finding an equivalent circuit that “gets rid of” the internal nodes between buses 1 and 2.

a) First, find the \(Y\)-bus matrix.

b) Schur complement can be used to partially solve linear equations. Namely, given \(M=\begin{bmatrix} A & B \\ D & C \end{bmatrix}\) and suppose we want to solve

\[\begin{bmatrix} A & B \\ D & C \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}.\]

Performing the operations used in Gaussian elimination, we can write \(x\) as

\[x = (M/C)^{-1} \hat{a}, \quad \hat{a}=a-BC^{-1}b.\]

Applying this to our circuit, and noticing that the current injections in the internal nodes all sum to $0$, find a relationship between just \(\begin{bmatrix} I_1 \\ I_2 \end{bmatrix}\) and \(\begin{bmatrix} V_1 \\ V_2 \end{bmatrix}\).

c) Draw the resulting equivalent circuit. Note this circuit might have shunt elements.