7.6 Powers, Discrete Dynamics, Markov Chains and Differential Equations

Module 7 — Eigenvalues and Dynamical Systems. Powers, Discrete Dynamics, Markov Chains and Differential Equations. Graded conceptual and computational practice with explained single-select and multiselect questions.

by Synapticz Media

Oct 20, 2026

16 quizzes

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16 quizzes

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7.6-01 Fast powers through diagonalization

7.6-01 Fast powers through diagonalization

Fast powers through diagonalization Use an eigenbasis to compute large matrix powers efficiently and understand why the formula works. Combine diagonal powers with basis changes, then apply the method to concrete two-by-two matrices and special spectra. What you will practise Cancel intermediate basis conversions Raise diagonal entries to powers Handle zero and negative exponents correctly Reconstruct an explicit power formula Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization and matrix inverses. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final structural items use powers of eigenvalues to identify matrix identities without repeated multiplication. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-02 Fibonacci as a matrix recurrence

7.6-02 Fibonacci as a matrix recurrence

Fibonacci as a matrix recurrence Model the Fibonacci recurrence as a two-component matrix update and recover its closed form from eigenmodes. The progression emphasizes state ordering, initial data and the distinction between a general recurrence and one specific sequence. What you will practise Construct the recurrence matrix Find its characteristic roots Match initial data to mode coefficients Interpret the dominant growth contribution Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review recurrences, diagonalization and quadratic roots. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. A final change of initial data shows how the same update matrix can produce a different closed-form sequence. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-03 Expanding an initial state

7.6-03 Expanding an initial state

Expanding an initial state Resolve an initial state into eigen-coordinates and evolve each component independently. Connect exact coordinate solving with decay, fixed modes, alternating behavior and absent growing modes. What you will practise Solve for initial mode coefficients Evolve and recombine the modes Find limits and subsequential limits Account for zero initial coefficients Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review eigenbases, small linear systems and powers. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final computation uses a nonorthogonal eigenbasis, making the initial coordinate conversion essential. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-04 Discrete growth and decay

7.6-04 Discrete growth and decay

Discrete growth and decay Predict long-run discrete behavior from eigenvalue moduli under explicit diagonalizability assumptions. Distinguish decay, growth, bounded oscillation and convergence, while accounting for the initial state. What you will practise Use modulus rather than sign alone Identify stable eigenspaces Separate boundedness from convergence Check complex eigenvalue magnitudes Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, eigenmodes and complex modulus. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The last case challenges reliance on the real part of a complex eigenvalue when the model advances in discrete steps. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-05 Phase portraits

7.6-05 Phase portraits

Phase portraits Classify planar discrete phase portraits from their eigenstructure. Identify sinks, sources, saddles and spirals, then compare the discrete interpretation with the continuous model using the same matrix. What you will practise Classify contracting and expanding modes Find stable directions of saddles Interpret turning and sign reversal State which time convention is used Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review discrete eigenmode evolution and complex conjugate pairs. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final question determines the exact set of initial states approaching a saddle rather than classifying every trajectory identically. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-06 Column-stochastic matrices

7.6-06 Column-stochastic matrices

Column-stochastic matrices Construct and use transition matrices with one consistent column-vector convention. Interpret individual entries, preserve probability mass, compose updates in the correct order and recognize absorbing states. What you will practise Build destination columns Check stochastic constraints Compute one-step distributions Compose transition matrices Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review probability vectors and matrix multiplication. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final example tracks remaining probability at a transient state next to an absorbing state. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-07 Stationary distributions

7.6-07 Stationary distributions

Stationary distributions Solve for stationary distributions by combining the fixed-vector equation with probability constraints. Compare unique, nonunique and periodic examples and derive a general two-state balance formula. What you will practise Solve stationary equations Normalize by total mass Check nonnegativity and uniqueness Separate stationary states from convergence Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review column-stochastic matrices, null spaces and linear systems. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The later problems examine convex mixtures, repeated transition columns and a parameterized two-state chain. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-08 Markov convergence and Perron–Frobenius

7.6-08 Markov convergence and Perron–Frobenius

Markov convergence and Perron–Frobenius Understand the conditions behind finite Markov-chain convergence. Use regularity, irreducibility and aperiodicity carefully, and examine counterexamples where a stationary distribution exists without convergence from every start. What you will practise Recognize positive matrix powers Distinguish uniqueness from attraction Diagnose periodic behavior Use self-loops and spectral information Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review stationary distributions, matrix powers and eigenvalue modulus. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final computation verifies regularity directly through a positive matrix square rather than through an unsupported shortcut. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-09 PageRank as a model

7.6-09 PageRank as a model

PageRank as a model Build a small PageRank model from link probabilities, a dangling-page rule and a teleportation distribution. Interpret the resulting stationary probabilities within the stated random-surfer model. What you will practise Create normalized outgoing-link columns Construct a damped transition matrix Handle pages without outgoing links Solve and interpret a stationary ranking Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review column-stochastic matrices and stationary distributions. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final two-page calculation combines damping and balance equations, while interpretation questions separate visitation from claims about information quality. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-10 Decoupling a differential system

7.6-10 Decoupling a differential system

Decoupling a differential system Decouple a constant-coefficient differential system using an eigenbasis. Convert the initial state, solve scalar exponential equations and reconstruct the solution in the original coordinates. What you will practise Derive the transformed differential system Convert initial conditions Evolve scalar exponential modes Recognize limitations of diagonal decoupling Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization, initial-value problems and scalar exponential derivatives. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final reasoning item explains why failure of an eigenbasis does not mean that the original differential system lacks solutions. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-11 Continuous-time stability

7.6-11 Continuous-time stability

Continuous-time stability Classify continuous-time linear stability using real parts of eigenvalues. Separate asymptotic decay, instability and bounded nondecaying motion, and pay special attention to defective boundary cases. What you will practise Apply the real-part criterion Identify sinks and saddles Treat imaginary-axis modes Analyze nilpotent and defective examples Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review continuous eigenmodes and basic matrix exponentials. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final comparison uses identical eigenvalue lists with different Jordan behavior to show why boundary stability requires more information. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-12 Trace–determinant portraits

7.6-12 Trace–determinant portraits

Trace–determinant portraits Locate real planar continuous systems in the trace–determinant classification. Use the discriminant to distinguish nodes and spirals, and inspect boundary cases where trace and determinant alone leave important structure unresolved. What you will practise Classify by trace and determinant Interpret the discriminant Identify centers and saddles Analyze a parameter family Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review quadratic characteristic equations and continuous-time stability. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final parameter example combines a fixed rotation rate with a variable real growth rate. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-13 Defining the matrix exponential

7.6-13 Defining the matrix exponential

Defining the matrix exponential Define the matrix exponential through its power series and verify its role as the solution operator for a constant linear differential system. Separate matrix powers from entrywise exponentiation and track the assumptions behind product identities. What you will practise Use the series definition Check the initial identity and derivative Establish finite-time invertibility Apply nilpotent truncation Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, differentiation and scalar exponential series. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final defective example computes an exact exponential without requiring diagonalization. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-14 Computing exponentials in an eigenbasis

7.6-14 Computing exponentials in an eigenbasis

Computing exponentials in an eigenbasis Compute matrix exponentials in an eigenbasis and apply them to initial conditions. Move from diagonal matrices to triangular and symmetric examples, using direct differentiation as a check on reconstructed formulas. What you will practise Exponentiate diagonal eigenvalues Transfer the exponential through similarity Apply initial-coordinate conversion Verify the resulting matrix function Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization and matrix exponential definitions. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final initial-value problem can be checked independently by solving its triangular differential equations. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-15 Rotation generators

7.6-15 Rotation generators

Rotation generators Derive rotations by exponentiating a real skew-symmetric planar generator. Connect alternating matrix powers to trigonometric series, interpret angular speed and add a commuting scalar growth or decay term. What you will practise Compute generator powers Recover sine and cosine from the series Determine direction and angular speed Prove conservation of Euclidean length Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review trigonometry, exponential series and differentiation. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final formula combines rotation with exponential damping using a justified commuting-factor identity. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.

7.6-16 Boundary cases and defective dynamics

7.6-16 Boundary cases and defective dynamics

Boundary cases and defective dynamics Examine the boundary cases where eigenvalue lists alone can mislead a dynamics calculation. Derive Jordan-block powers and compare transient growth, eventual decay and continuous nilpotent evolution. What you will practise Detect polynomial growth at unit modulus Account for the initial state Separate transient amplification from long-run behavior Compare discrete powers with continuous exponentials Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix powers, defective eigenspaces and basic exponentials. The questions move from recognition and short calculations to connected reasoning. Single-select questions require one answer; multiselect questions ask you to evaluate every statement independently. The final comparison uses one nilpotent matrix in two different time models and explains why their trajectories behave differently. Work out your own result before checking the choices, then use the explanation to compare the reasoning and diagnose a mistaken assumption. This focused practice connects conceptual understanding with reliable problem solving and supports review before the next microlearning.