Is the following statement true? Equivalent matrices are similar

Is this true?

Equivalent matrices are similar

No — it is false.

The counterexample

$A = I_{2}$ and $B = 2 I_{2}$

Both are invertible, so each is P·(the other)·Q for suitable invertible P, Q — every invertible n×n matrix is equivalent to every other, because equivalence sees nothing finer than rank. But A has eigenvalue 1 and B has eigenvalue 2, and similarity preserves eigenvalues. Equivalence allows an independent change of basis in the domain and the codomain; similarity forces the same basis on both sides.

The kind of mistake this is

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

See real exam options built on this mistake Drill statements like this

linear algebra change of basis

Others that fail the same way

Same characteristic polynomial ⇒ similar
Diagonalisable ⇒ invertible
Real matrix with real eigenvalues is diagonalisable
AB = I ⇒ BA = I for all matrices
Every real symmetric matrix is positive definite if det > 0
Same characteristic and minimal polynomial ⇒ similar

Explained in full in Linear transformations, matrices and similarity

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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