7.4 Diagonalization

Module 7 — Eigenvalues and Dynamical Systems. Diagonalization. Graded conceptual and computational practice with explained single-select and multiselect questions.

by Synapticz Media

Oct 20, 2026

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7.4-01 Constructing an eigenbasis

Constructing an eigenbasis Build an eigenbasis by combining independent directions from the relevant eigenspaces. The set emphasizes spanning the entire space, repeated eigenvalues and the choice between real and complex scalars. What you will practise Test independence of eigenvectors Assemble bases from distinct eigenspaces Use dimension counts to detect a deficit Specify the field for an eigenbasis Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review bases, eigenspaces and determinants. 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 selection task compares several candidate bases for a matrix with a repeated eigenvalue. 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.

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7.4-02 Building A = PDP⁻¹

Building A = PDP⁻¹ Construct and verify a diagonalization with consistent eigenvector and eigenvalue ordering. Interpret the change-of-basis matrices as coordinate conversions and practise reconstructing the original matrix. What you will practise Build the eigenvector matrix Match diagonal entries to columns Verify the product identity Track valid changes to an eigenbasis Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review matrix multiplication, inverses and eigenbases. 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. Later questions distinguish harmless basis changes inside an eigenspace from changes that destroy independence or the eigenvector property. 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.

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7.4-03 Independence of distinct-eigenvalue vectors

Independence of distinct-eigenvalue vectors Understand why eigenvectors associated with distinct eigenvalues are independent, and what this theorem does not say. Computation, counterexamples and short proof steps connect the theorem to diagonalization. What you will practise Apply the independence theorem Separate independence from orthogonality Isolate coefficients using shifted operators Recognize the limits of the converse Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review linear dependence and eigenvector equations. 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 problem examines sums across distinct eigenspaces and explains why they normally stop being eigenvectors. 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.

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7.4-04 Algebraic and geometric multiplicity

Algebraic and geometric multiplicity Compare algebraic multiplicity, which comes from a characteristic polynomial, with geometric multiplicity, which comes from a null space. Use scalar matrices, defective blocks and rank data to see both equality and deficiency. What you will practise Read algebraic multiplicity Compute geometric multiplicity Apply multiplicity bounds Count the available independent eigenvectors Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review polynomial factors, kernels and rank-nullity. 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 examples combine repeated-root data with shifted ranks to decide whether enough independent eigenvectors exist. 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.

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7.4-05 Testing diagonalizability

Testing diagonalizability Test diagonalizability over the stated field using eigenbasis and eigenspace-dimension criteria. Examples include repeated roots, singular matrices, complex rotations and matrices satisfying simple polynomial identities. What you will practise Apply the eigenbasis criterion Check splitting and multiplicity equality Distinguish diagonalization from invertibility Use polynomial identities to reason about structure Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review eigenspaces, algebraic multiplicity and bases. 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 problems use special matrix identities to derive stronger conclusions than a simple root count provides. 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.

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7.4-06 Defective and parameter-dependent cases

Defective and parameter-dependent cases Study parameter values where eigenvalues collide or an eigenspace changes dimension. The examples compare harmless repetition with genuine defects and include a family whose real and complex diagonalizability differ. What you will practise Identify exceptional parameter values Solve shifted systems at collisions Separate repeated roots from missing eigenvectors Track field-dependent cases Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review characteristic polynomials, null spaces and diagonalizability criteria. 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 three-dimensional example shows how coupling through another coordinate can reduce a repeated eigenspace. 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.

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7.4-07 Similarity invariants

Similarity invariants Explain similarity as a change of coordinates for one transformation. Establish its spectral invariants and explore why equal trace, determinant or even characteristic polynomial need not establish similarity. What you will practise Transfer eigenvectors between bases Prove characteristic-polynomial invariance Compare multiplicities and ranks Construct counterexamples to incomplete tests Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review change of basis, determinants and eigenspaces. 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. Later questions distinguish spectral invariants from Euclidean properties that a general nonorthogonal basis change can alter. 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.

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7.4-08 Commuting and simultaneous diagonalization

Commuting and simultaneous diagonalization Determine when two transformations admit one common eigenbasis. Derive commutation from simultaneous diagonalization and examine the additional hypotheses needed to reverse that implication. What you will practise Define a shared diagonalization Use commutation to preserve eigenspaces Apply the simple-spectrum argument Check concrete commuting and noncommuting pairs Prerequisites and study approach This microlearning belongs to Module 7, Eigenvalues and Dynamical Systems, in the Complete Linear Algebra Course. Before starting, review diagonalization, matrix products and eigenspaces. 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 pair is individually diagonalizable but fails the necessary product test for a shared 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.

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