6.3-02 Idempotence and symmetry

Idempotence and symmetry

Idempotence and symmetry explain the algebra of orthogonal projection. The questions distinguish those two properties from invertibility and orthogonality of a square matrix, using concrete tests and short structural arguments.

What you will practise

• Interpret repeated projection.
• Check symmetry and idempotence separately.
• Use range, kernel and energy identities.
• Investigate allowable eigenvalues and boundary cases.

Begin with recognition and interpretation, then work through calculations, error diagnosis and more demanding reasoning. Single-select and multi-select questions ask you to evaluate the stated assumptions as well as the result. Worked explanations connect the answer to a method or a counterexample, so you can check why tempting alternatives fail.

Before you start

Review matrix multiplication and orthogonal decomposition. Keep a written calculation or short justification for each response, especially when several statements can be correct. This microlearning supports recognizing projectors and distinguishing oblique projection within Module 6 of the Complete Linear Algebra Course.