Compute eigenspaces and distinguish zero eigenvalues from forbidden zero eigenvectors.
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Try
The diagonal map diag(3,−1) stretches the horizontal direction threefold and reverses the vertical direction. Those two directions are preserved as lines. An eigenvector v is nonzero and satisfies Av=λv for a scalar eigenvalue λ.
Predict
Predict whether λ=0 is allowed. It is: any nonzero kernel vector is an eigenvector with eigenvalue zero. The zero vector itself is excluded because its equation holds for every λ and identifies no special direction. An eigenspace includes zero along with all eigenvectors for one fixed λ.
Experiment
For A=[[2,1],[0,3]], compute A(1,0) and A(1,1). Both outputs are multiples of their inputs, with factors 2 and 3. Try (0,1); its output (1,3) changes direction. Record the difference between a preserved line and a generally moved vector.
Equal-scale eigenvector comparison for A=[[2,1],[0,3]]. The direction (1,0) is scaled by 2; (1,1) is scaled by 3. Input and output endpoints are labeled separately.
Understand
Worked computation: (A−λI)v=0 has a nonzero solution exactly when det(A−λI)=0. Here the determinant is (2−λ)(3−λ), yielding λ=2 or 3. At 2 the equations force y=0; at 3 they force y=x. Thus the eigenspaces are span((1,0)) and span((1,1)). Finding roots is only the first step; one must solve the corresponding kernels.
The characteristic polynomial has degree n for an n×n matrix. Over the reals it may have nonreal roots: a 90° plane rotation has equation λ²+1=0 and no real eigenvector direction. Complex extensions belong to the later theory.
Invalid shortcut: the diagonal entries of every matrix are its eigenvalues. That holds for triangular matrices, not arbitrary ones.
Apply
Check each candidate by direct multiplication and confirm it is nonzero. State the scalar field when describing missing eigenvalues. Distinguish the algebraic multiplicity of a polynomial root from the dimension of its eigenspace.
Continue
Depth: eigenvectors associated with distinct eigenvalues are independent. Repeated roots need additional analysis, which motivates the next lesson. Eigenvalues alone also do not determine the sensitivity of a numerical computation.
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A=diag(4,−2). For v=(0,3), find λ such that Av=λv.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A 2×2 matrix has characteristic polynomial λ²−7λ+10. If one eigenvalue is 2, find the other.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Why is zero excluded from the definition of an eigenvector?
The real 90° rotation matrix [[0,−1],[1,0]] has:
Find all eigenvalues and eigenspaces of A=[[2,1],[0,3]].
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For fixed real λ, prove the eigenspace ker(A−λI) is a subspace and explain when it contains an eigenvector.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
