Linear algebra MOC

Trace

The trace tr⁡𝑇 of a linear endomorphism 𝑇 :𝑉 →𝑉 is the sum of its eigenvalues counted with (algebraic) multiplicity. linalg This corresponds to the sum of main diagonal entries in any basis.

Properties

• If 𝐀 and 𝐁 are real/complex 𝑛 ×𝑛 matrices, then tr⁡𝐀𝐁 =tr⁡𝐀tr⁡𝐁.
• If 𝐀 is nilpotent, tr⁡𝐀 =0.
• tr⁡𝐀𝐁 =tr⁡𝐁𝐀
• tr⁡𝐓−1𝐀𝐓 =tr⁡𝐀 for invertible 𝐓

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