CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-11

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Similar matrices have the same trace

Statement

If square matrices $A$ and $B$ over $F$ are similar, then $tr (A) = tr (B)$ .

Facts & Assumptions

Given: Similar matrices $A, B \in M_{n} (F)$ .

[L1]

Similarity means $B = P^{- 1} A P$ for an invertible matrix $P$ (Similar matrices: $B = P^{- 1} A P$ for an invertible $P$ ).

[L2]

For conformable rectangular matrices $X, Y$ , one has $tr (X Y) = tr (Y X)$ (For $A \in M_{m \times n} (F)$ and $B \in M_{n \times m} (F)$ , $tr (A B) = tr (B A)$ ).

[L3]

Matrix multiplication is associative and unital (Matrix multiplication is associative, unital, distributive, and compatible with scalar multiplication).

Proof

technique · direct

1.1

Choose invertible $P$ with $B = P^{- 1} A P$ as in [L1].

givenL1

2.1

Apply [L2] to $X = P^{- 1} A$ and $Y = P$ to obtain $tr (B) = tr (P (P^{- 1} A))$ .

step 1.1L1L2

3.1

By associativity and unitality from [L3], $P (P^{- 1} A) = (P P^{- 1}) A = I_{n} A = A$ , so $tr (B) = tr (A)$ .

step 2.1L1L2L3∎

Depends on

Used by

Matrix coefficients and characters Definition
The basis-independent trace of an endomorphism of a finite-dimensional vector space Definition
Real Cartan subalgebras need not be conjugate Proposition
A finite simple graph is bipartite if and only if its adjacency spectrum is symmetric about 0 Theorem
Restricted root space decomposition Theorem
Schur orthogonality Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Axler, Linear Algebra Done Right, 4th ed., Theorem 8.50 (standard reference, not scraped)