Linear algebra in engineering: case studies and a project

Linear algebra is the computational backbone of modern engineering. Structural analysis, circuit simulation, control design, signal processing, computer graphics, machine learning and the finite element method all reduce, at their core, to vectors, matrices, linear systems and eigenvalue problems. This course has built the tools: vectors and their combinations (Lessons 1 and 2), linear systems and their efficient solution (Lessons 3 and 5), matrix algebra and determinants (Lessons 4 and 6), the structure of solutions (Lesson 7), transformations (Lesson 8), eigenvalues (Lessons 9 and 11), and least squares (Lesson 10). This final lesson applies them together in three case studies, each typical of a different engineering discipline, and then sets a course project. The emphasis is on formulating problems in matrix form, which is the step that software cannot do for you, and on interpreting the results.

Case study 1: forces in a truss

A truss carries loads through members that are in pure tension or compression. Its analysis is a linear system: at each joint, the member forces and external loads must balance in both directions. Consider a triangular truss with joints A at (0, 0), B at (4, 0) and C at (2, 2) meters, members AB, AC and BC, a pin support at A (reactions \(A_x\), \(A_y\)), a roller at B (reaction \(B_y\)), and a 10 kN downward load at C. Member forces \(F_{AB}\), \(F_{AC}\), \(F_{BC}\) are taken positive in tension.

• Each member force acts along the member, so it contributes its force times a unit vector to each joint it connects. The unit vector from C toward A is \((-1, -1)/\sqrt{2}\), from C toward B is \((1, -1)/\sqrt{2}\), and from A toward B is \((1, 0)\).
• Three joints give six equations in six unknowns (three member forces and three reactions), so the system is square, matching the count of Lesson 7 for a determinate truss.

In the stiffness method used by structural software, the unknowns are the joint displacements instead of the forces. Each member contributes a small symmetric stiffness matrix, and these are assembled into a global stiffness matrix \(K\), which is symmetric, sparse and, for a properly supported structure, positive definite. The structure's response to loads \(\mathbf{f}\) is found by solving \(K\mathbf{u} = \mathbf{f}\), usually by Cholesky factorization, and its natural vibrations from the eigenvalue problem \(K\boldsymbol{\phi} = \omega^2M\boldsymbol{\phi}\) of Lesson 11. The finite element method extends the same assembly process to plates, shells and solids with millions of unknowns.

Case study 2: nodal analysis of a circuit

Circuit simulators write Kirchhoff's current law at each node in terms of the unknown node voltages, giving a linear system \(G\mathbf{v} = \mathbf{i}\), where \(G\) is the conductance matrix and \(\mathbf{i}\) holds the source currents injected at each node. Each conductance \(1/R\) connecting two nodes adds to both their diagonal entries and subtracts from the two corresponding off-diagonal entries, so \(G\) is symmetric; a conductance to ground adds only to the diagonal.

Case study 3: reliability as a Markov chain

Many systems move randomly between a few states: a machine may be running, degraded or failed; a traffic signal's queue may be short, medium or long; a customer may be with one supplier or another. A Markov chain models such a system with a transition matrix \(P\) whose entry \(p_{ij}\) is the probability of moving from state \(j\) to state \(i\) in one time step. Each column sums to 1. If \(\mathbf{x}_k\) holds the probabilities of being in each state at step \(k\), then \(\mathbf{x}_{k+1} = P\mathbf{x}_k\), and \(\mathbf{x}_k = P^k\mathbf{x}_0\), a matrix power of the kind computed with eigenvalues in Lesson 9.

A transition matrix always has the eigenvalue 1, and for most practical chains its eigenvector, scaled so its entries sum to 1, is the steady-state distribution that the system approaches regardless of where it starts. The other eigenvalues, all smaller than 1 in magnitude, determine how quickly the approach happens.

What the case studies share

In each case study, the engineering content lay in setting up the matrix: choosing unknowns, writing one equation per balance law (forces at a joint, currents at a node, probabilities into a state), and recognizing the structure of the resulting matrix. Once set up, the computations used standard tools whose properties this course has explained: symmetric positive definite systems solved reliably by factorization, ranks that confirm a model is well posed, and eigenvectors that describe long-run or natural behavior. Recognizing these structures tells an engineer which software method to choose and what could go wrong.

Linear algebra and computation

In practice, engineers carry out these calculations with numerical software: MATLAB, Python with the NumPy and SciPy libraries, Julia, spreadsheets for small problems, and specialized programs for structures, circuits and fluids. All of them call the same core numerical libraries, developed over decades, that implement the LU, Cholesky, QR, eigenvalue and singular value algorithms described in this course. Using them well requires understanding what they compute: that solving is better than inverting, that symmetric positive definite matrices deserve a Cholesky solver, that a warning about a near-singular matrix signals a modeling problem, and that the condition number limits the accuracy of any answer. ENCM-100 introduces programming for these computations.

The course project

Choose one of the following problems, or propose your own of similar scope with your instructor's approval, and prepare a technical report with your formulation, calculations (by hand for small cases and with software for larger ones), results and interpretation.

• Project A: Truss analysis. Design a planar truss with at least seven members for a stated span and load. Assemble the equilibrium matrix, check its rank for stability and determinacy, solve for the member forces, and identify the most heavily loaded members. Then add one member and describe the null space of self-stresses that appears.
• Project B: Circuit or pipe network. Analyze a resistive circuit with at least four nodes, or a water distribution network linearized around an operating point, by nodal analysis. Show that the matrix is symmetric and positive definite, solve it by Cholesky or Gauss-Seidel, and check the solution against Kirchhoff's laws.
• Project C: Data fitting. Collect or obtain a data set from an engineering source (a calibration, a material test, a temperature record). Fit at least two candidate models by least squares, compare their residuals, check for overfitting by holding back some data, and discuss which model you would use and why.
• Project D: Vibration or Markov model. Model a multi-story frame, a vehicle suspension or a chain of masses and springs, and find its natural frequencies and mode shapes; or model a system that moves between states (equipment condition, traffic, inventory) as a Markov chain, and find its steady state and how fast it is reached.

Every report should define the unknowns and assumptions, show how each equation follows from a physical law, state the size, structure and rank of the matrices, verify the solution independently, discuss sensitivity to the most uncertain inputs, and interpret the results in engineering terms.

Looking ahead

Linear algebra continues throughout an engineering program. Multivariable calculus (MATH-209) uses vectors and matrices to describe gradients, Jacobians and coordinate changes. Differential equations (MATH-201) solves systems of equations using the eigenvalues and eigenvectors of Lesson 9, and numerical methods (MATH-300) develops the algorithms of Lessons 5 and 10 further. Structural analysis, the finite element method, circuit and power system analysis, control systems, robotics and machine learning all build directly on this course.