Hidden planes, compatible metrics and canonical normalization
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 150 minutes, as an unvalidated planning estimate. This collection contains 3 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Work in finite-dimensional real inner-product spaces unless complex scalars are explicitly introduced. Complex inner products are linear in their first argument. State which metric defines every adjoint.
These three original problems are public supported proof practice. The suggested time is an author estimate; saved explanations and disclosed worked solutions do not constitute independent assessment.
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On Euclidean R⁴ let S have rows (0,−3,−4,0),(3,0,0,0),(4,0,0,0),(0,0,0,0). Construct an explicit orthonormal basis displaying its nonzero rotation plane and entire kernel. Verify the block matrix by images of basis vectors and compute the characteristic polynomial. Derive exp(tS)e₂ in original coordinates for every real t, checking its initial value and constant norm. Explain why S cannot be diagonalized over R, identify complex eigenvectors, and distinguish this from the diagonalizability of −S².
Structure your argument and preserve revisions
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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.
Let J=((1,−2),(1,−1)) act on real column vectors. Verify J²=−I but show J is not skew-adjoint for the Euclidean inner product. Classify every real symmetric positive-definite matrix G for which JᵀG+GJ=0. For the member with G₁₁=1, construct a complex coordinate map Φ:R²→C that intertwines J with multiplication by i. Verify directly that h(x,y)=xᵀGy−i(Jx)ᵀGy equals Φ(x)conj(Φ(y)), and explain the norm preserved by exp(tJ). Use a specific vector to show Euclidean length need not be preserved.
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.
Let S be skew-adjoint on a finite-dimensional real inner-product space V. Put H=(ker S)⊥ and assume H is nonzero. On H let R be the positive-definite self-adjoint square root of −S². You may use existence and uniqueness of a positive square root, but must justify commutation wherever needed. Prove J=SR⁻¹ is an orthogonal complex structure on H. Prove it is the unique skew-adjoint K on H with K²=−I, KS=SK and −KS positive definite. Then analyze S=diag(3J₀,7J₀,0₂) on Euclidean R⁶, where J₀=(0,−1;1,0): give two distinct commuting orthogonal extensions of J to V, explain why strict positivity cannot hold on V, and decide what changes if the kernel has dimension one instead of two.
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
On Euclidean R⁴ let S have rows (0,−3,−4,0),(3,0,0,0),(4,0,0,0),(0,0,0,0). Construct an explicit orthonormal basis displaying its nonzero rotation plane and entire kernel. Verify the block matrix by images of basis vectors and compute the characteristic polynomial. Derive exp(tS)e₂ in original coordinates for every real t, checking its initial value and constant norm. Explain why S cannot be diagonalized over R, identify complex eigenvectors, and distinguish this from the diagonalizability of −S².
Let J=((1,−2),(1,−1)) act on real column vectors. Verify J²=−I but show J is not skew-adjoint for the Euclidean inner product. Classify every real symmetric positive-definite matrix G for which JᵀG+GJ=0. For the member with G₁₁=1, construct a complex coordinate map Φ:R²→C that intertwines J with multiplication by i. Verify directly that h(x,y)=xᵀGy−i(Jx)ᵀGy equals Φ(x)conj(Φ(y)), and explain the norm preserved by exp(tJ). Use a specific vector to show Euclidean length need not be preserved.
Let S be skew-adjoint on a finite-dimensional real inner-product space V. Put H=(ker S)⊥ and assume H is nonzero. On H let R be the positive-definite self-adjoint square root of −S². You may use existence and uniqueness of a positive square root, but must justify commutation wherever needed. Prove J=SR⁻¹ is an orthogonal complex structure on H. Prove it is the unique skew-adjoint K on H with K²=−I, KS=SK and −KS positive definite. Then analyze S=diag(3J₀,7J₀,0₂) on Euclidean R⁶, where J₀=(0,−1;1,0): give two distinct commuting orthogonal extensions of J to V, explain why strict positivity cannot hold on V, and decide what changes if the kernel has dimension one instead of two.
