Matrix Similarity Calculator
Determine if two matrices are similar, find similarity transformations, and analyze matrix equivalence relationships. Check eigenvalues, find diagonalization, and understand the geometric interpretation of matrix similarity with detailed step-by-step solutions.
Matrix Input
Matrix Similarity Theory
What is Matrix Similarity?
Two square matrices A and B are similar if there exists an invertible matrix P such that B = P⁻¹AP. This relationship means the matrices represent the same linear transformation under different bases. Similar matrices share many important properties including eigenvalues, determinant, trace, and rank.
Similarity Definition:
A ~ B ⟺ ∃P invertible: B = P⁻¹AP
Equivalently: AP = PB
Properties preserved:
• Same eigenvalues (including multiplicities)
• Same determinant: det(A) = det(B)
• Same trace: tr(A) = tr(B)
• Same rank: rank(A) = rank(B)
• Same characteristic polynomial
How to Check Similarity
Step 1: Check Necessary Conditions
Verify that det(A) = det(B), tr(A) = tr(B), and rank(A) = rank(B)
Step 2: Compare Eigenvalues
Calculate eigenvalues of both matrices - they must be identical
Step 3: Check Diagonalizability
If both matrices are diagonalizable with same eigenvalues, they're similar
Step 4: Find Transformation Matrix
If similar, find P such that B = P⁻¹AP using eigenvectors
Types of Matrix Similarity
Diagonal Similarity
A ~ D where D is diagonal
Possible when A has n linearly independent eigenvectors
Jordan Similarity
A ~ J where J is in Jordan normal form
Always possible over algebraically closed fields
Orthogonal Similarity
A ~ B with P orthogonal (P⁻¹ = Pᵀ)
Preserves lengths and angles
Unitary Similarity
A ~ B with P unitary (P⁻¹ = P*)
Complex analog of orthogonal similarity
Applications
Linear Algebra
Change of basis, canonical forms, matrix classification
Differential Equations
System solutions, stability analysis, normal modes
Quantum Mechanics
Unitary transformations, observable operators
Computer Graphics
Coordinate transformations, rotations, scaling
Data Analysis
Principal component analysis, dimensionality reduction
Control Theory
System equivalence, state space transformations
Worked Examples
Example 1: Similar 2×2 Matrices
Problem:
Check if matrices A and B are similar:
[0 2] [0 1]
Solution:
Step 1: Check basic properties
det(A) = 6, det(B) = 4 ≠ 6
Step 2: Since determinants differ, matrices are not similar
Result: A and B are not similar
Example 2: Diagonal Similarity
Problem:
Show that A is similar to a diagonal matrix:
[0 2]
Solution:
Step 1: Find eigenvalues: λ₁ = 1, λ₂ = 2
Step 2: Find eigenvectors: v₁ = [1,0]ᵀ, v₂ = [1,1]ᵀ
Step 3: P = [1 1; 0 1], D = [1 0; 0 2]
Step 4: Verify: A = PDP⁻¹
Result: A ~ D with transformation matrix P
Example 3: 3×3 Similarity Check
Problem:
Determine if these 3×3 matrices are similar:
[0 2 1] [1 2 0]
[0 0 2] [0 1 2]
Solution:
Step 1: Both have eigenvalue λ = 2 with multiplicity 3
Step 2: Check Jordan forms - both have same Jordan structure
Step 3: Same characteristic and minimal polynomials
Result: Matrices are similar (both similar to same Jordan form)
Frequently Asked Questions
Similar matrices are related by B = P⁻¹AP (same linear transformation, different bases). Equivalent matrices are related by row/column operations. Similarity is stronger - similar matrices are always equivalent, but not vice versa.
Yes! Similar matrices always have identical eigenvalues (including multiplicities). This is because similarity preserves the characteristic polynomial. However, having the same eigenvalues doesn't guarantee similarity.
If both matrices are diagonalizable, find their eigenvector matrices P_A and P_B. Then P = P_B P_A⁻¹. For general cases, you may need to work with Jordan forms or solve the equation AP = PB directly.
Yes! Non-diagonalizable matrices can be similar if they have the same Jordan normal form. The Jordan form is the canonical form for similarity - two matrices are similar if and only if they have the same Jordan form.
Similar matrices represent the same linear transformation expressed in different coordinate systems. The transformation P represents a change of basis - similarity shows that the "essential behavior" of the transformation is preserved regardless of the coordinate system chosen.
No! Having the same eigenvalues is necessary but not sufficient. Matrices must also have the same geometric multiplicities for each eigenvalue. For example, [2 1; 0 2] and [2 0; 0 2] both have eigenvalue 2 with algebraic multiplicity 2, but different geometric multiplicities.