Two matrices are congruent if one can be obtained from the other by pre- and post-multiplication by the transpose of an invertible matrix.
⚠For symmetric matrices, Sylvester's Law of Inertia states: two symmetric matrices are congruent iff they have same rank and same numbers of positive, negative, and zero eigenvalues.
What are Congruent Matrices?
Two matrices A and B are congruent if there exists an invertible matrix P such that B = PᵀAP. Congruent matrices represent the same bilinear form under different coordinate systems.
Bilinear Forms
Congruent matrices represent same bilinear form in different bases.
Inertia Preserved
Numbers of positive, negative, zero eigenvalues stay same.
Symmetric Case
Symmetric matrices congruent iff same rank and inertia.
vs Similarity
Similar: B = P⁻¹AP. Congruent: B = PᵀAP.
Teaching Example:
A = [[2,1],[1,3]], B = [[1,0],[0,6]]
1. det(A) = 2*3 - 1*1 = 5 > 0
2. det(B) = 1*6 = 6 > 0
3. Both 2x2 with det > 0
4. A ≃ B (same positive definiteness)
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