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< 14.5 Matrix Inversion | Contents | CHAPTER 15. Eigenvalues and Eigenvectors >

# Summary¶

1. Linear algebra is the foundation of many engineering fields.

2. Vectors can be considered as points in $${\mathbb{R}}^n$$; addition and multiplication are defined on them, although not necessarily the same as for scalars.

3. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others.

4. Matrices are tables of numbers. They have several important properties including the determinant, rank, and inverse.

5. A system of linear equations can be represented by the matrix equation $$Ax = y$$.

6. The number of solutions to a system of linear equations is related to the rank($$A$$) and the rank($$[A,y]$$). It can be zero, one, or infinity.

7. We can solve the equations using Gauss Elimination, Gauss-Jordan Elimination, LU decomposition and Gauss-Seidel method.

8. We introduced methods to find matrix inversion.

# Problems¶

1. Show that matrix multiplication distributes over matrix addition: show $$A(B + C) = AB + AC$$ assuming that $$A, B$$, and $$C$$ are matrices of compatible size.

2. Write a function my_is_orthogonal(v1,v2, tol), where $$v1$$ and $$v2$$ are column vectors of the same size and $$tol$$ is a scalar value strictly larger than 0. The output should be 1 if the angle between $$v1$$ and $$v2$$ is within tol of $$\pi/2$$; that is, $$|\pi/2 - \theta| < \text{tol}$$, and 0 otherwise. You may assume that $$v1$$ and $$v2$$ are column vectors of the same size, and that $$tol$$ is a positive scalar.

# Test cases for problem 2
a = np.array([[1], [0.001]])
b = np.array([[0.001], [1]])
# output: 1
my_is_orthogonal(a,b, 0.01)

# output: 0
my_is_orthogonal(a,b, 0.001)

# output: 0
a = np.array([[1], [0.001]])
b = np.array([[1], [1]])
my_is_orthogonal(a,b, 0.01)

# output: 1
a = np.array([[1], [1]])
b = np.array([[-1], [1]])
my_is_orthogonal(a,b, 1e-10)

1. Write a function my_is_similar(s1,s2,tol) where $$s1$$ and $$s2$$ are strings, not necessarily the same size, and $$tol$$ is a scalar value strictly larger than 0. From $$s1$$ and $$s2$$, the function should construct two vectors, $$v1$$ and $$v2$$, where $$v1[0]$$ is the number of ‘a’s in $$s1$$, $$v1[1]$$ is the number ‘b’s in $$s1$$, and so on until $$v1[25]$$, which is the number of ‘z’s in $$v1$$. The vector $$v2$$ should be similarly constructed from $$s2$$. The output should be 1 if the absolute value of the angle between $$v1$$ and $$v2$$ is less than tol; that is, $$|\theta| < \text{tol}$$.

1. Write a function my_make_lin_ind(A), where $${A}$$ and $${B}$$ are matrices. Let the $${rank(A) = n}$$. Then $${B}$$ should be a matrix containing the first $$n$$ columns of $${A}$$ that are all linearly independent. Note that this implies that $${B}$$ is full rank.

## Test cases for problem 4

A = np.array([[12,24,0,11,-24,18,15],
[19,38,0,10,-31,25,9],
[1,2,0,21,-5,3,20],
[6,12,0,13,-10,8,5],
[22,44,0,2,-12,17,23]])

B = my_make_lin_ind(A)

# B = [[12,11,-24,15],
#      [19,10,-31,9],
#      [1,21,-5,20],
#      [6,13,-10,5],
#      [22,2,-12,23]]

1. Cramer’s rule is a method of computing the determinant of a matrix. Consider an $${n} \times {n}$$ square matrix $$M$$. Let $$M(i,j)$$ be the element of $$M$$ in the $$i$$-th row and $$j$$-th column of $$M$$, and let $$m_{i,j}$$ be the minor of $$M$$ created by removing the $$i$$-th row and $$j$$-th column from $$M$$. Cramer’s rule says that $$$\text{det(M)} = \sum_{i = 1}^{n} (-1)^{i-1} M(1,i) \text{det}(m_{i,j}).$$$

Write a function my_rec_det(M), where the output is $$det(M)$$. The function should use Cramer’s rule to compute the determinant, not Numpy’s function.

1. What is the complexity of my_rec_det in the previous problem? Do you think this is an effective way of determining if a matrix is singular or not?

2. Let $${p}$$ be a vector with length $${L}$$ containing the coefficients of a polynomial of order $${L-1}$$. For example, the vector $${p = [1,0,2]}$$ is a representation of the polynomial $$f(x) = 1 x^2 + 0 x + 2$$. Write a function my_poly_der_mat(p), where $${p}$$ is the aforementioned vector, and the output $$D$$ is the matrix that will return the coefficients of the derivative of $${p}$$ when $${p}$$ is left multiplied by $${D}$$. For example, the derivative of $$f(x)$$ is $$f'(x) = 2x$$, and therefore, $$d = Dp$$ should yield $${d = [2,0]}$$. Note this implies that the dimension of $$D$$ is $${L-1} \times {L}$$. The point of this problem is to show that integrating polynomials is actually a linear transformation.

1. Use Gauss Elimination to solve the following equations.

$\begin{eqnarray*} 3x_1 - x_2 + 4x_3 &=& 2 \\ 17x_1 + 2x_2 + x_3 &=& 14 \\ x_1 + 12x_2 -7z &=& 54 \\ \end{eqnarray*}$
1. Use Gauss-Jordan Elimination to solve the above equations in problem 8.

2. Get the lower triangular matrix $$L$$ and upper triangular matrix $$U$$ from the equations in problem 8.

3. Show that the dot product distributes across vector addition, that is, show that $$u \cdot (v + w) = u \cdot v + u \cdot w$$.

4. Consider the following network consisting of two power supply stations denoted by $$S1$$ and $$S2$$ and five power recipient nodes denoted by $$N1$$ to $$N5$$. The nodes are connected by power lines, which are denoted by arrows, and power can flow between nodes along these lines in both directions.

Let $$d_{i}$$ be a positive scalar denoting the power demands for node $$i$$, and assume that this demand must be met exactly. The capacity of the power supply stations is denoted by $$S$$. Power supply stations must run at their capacity. For each arrow, let $$f_{j}$$ be the power flow along that arrow. Negative flow implies that power is running in the opposite direction of the arrow.

Write a function my_flow_calculator(S, d), where $$S$$ is a $$1 \times 2$$ vector representing the capacity of each power supply station, and $$d$$ is a $$1 \times 5$$ row vector representing the demands at each node (i.e., $$d[0]$$ is the demand at node 1). The output argument, $$f$$, should be a $$1 \times 7$$ row vector denoting the flows in the network (i.e., $$f[0] = f_1$$ in the diagram). The flows contained in $$f$$ should satisfy all constraints of the system, like power generation and demands. Note that there may be more than one solution to the system of equations.

The total flow into a node must equal the total flow out of the node plus the demand; that is, for each node $$i, f_{\text{inflow}} = f_{\text{outflow}} + d_{i}$$. You may assume that $$\Sigma{S_j} = \Sigma{d_i}$$.

## Test cases for problem 4

s = np.array([[10, 10]])
d = np.array([[4, 4, 4, 4, 4]])

# f = [[10.0, 4.0, -2.0, 4.5, 5.5, 2.5, 1.5]]
f = my_flow_calculator(s, d)

s = np.array([[10, 10]])
d = np.array([[3, 4, 5, 4, 4]])
# f = [[10.0, 5.0, -1.0, 4.5, 5.5, 2.5, 1.5]]
f = my_flow_calculator(s, d)


< 14.5 Matrix Inversion | Contents | CHAPTER 15. Eigenvalues and Eigenvectors >