Two matrices are similar if they have the same Jordan canonical form. For diagonalizable matrices, same eigenvalues guarantee similarity.
⚠Same trace and determinant are necessary but NOT sufficient for similarity. Need same Jordan form structure.
What are Similar Matrices?
Two matrices A and B are similar if there exists an invertible matrix P such that B = P⁻¹AP. Similar matrices represent the same linear transformation with respect to different bases.
Invariant Properties
Trace, determinant, eigenvalues, rank are preserved under similarity.
Jordan Form
Matrices are similar iff they have identical Jordan canonical forms.
Diagonalizable
Diagonalizable matrices are similar iff they have same eigenvalues.
Transformation
Similar matrices represent same linear transformation in different bases.
Teaching Example:
A = [[1,2],[3,4]], B = [[5,-2],[-3,0]]
1. tr(A) = 5, tr(B) = 5 ✓
2. det(A) = -2, det(B) = -2 ✓
3. Same characteristic polynomial ✓
4. A ~ B (they are similar)
Applications
Linear TransformationsChange of BasisJordan FormMatrix DiagonalizationControl Theory
Frequently Asked Questions
What are similar matrices?▼
A ~ B if exists invertible P such that B = P⁻¹AP. Similar matrices share eigenvalues, trace, determinant, rank.
How to check similarity?▼
Necessary conditions: same eigenvalues, trace, determinant, rank. Sufficient: same Jordan canonical form.
Similar vs congruent?▼
Similar: B = P⁻¹AP. Congruent: B = PᵀAP. Different relations.
Diagonalizable similar matrices?▼
Two diagonalizable matrices are similar iff they have same eigenvalues with same multiplicities.
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