The discussion centers on proving that every involutory matrix can be expressed as a difference of two idempotents. An involutory matrix, defined by the property that ##A^2 = I##, can be decomposed using the formula ##A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##. The challenge lies in demonstrating that both matrices ##\frac{1}{2}(I+A)## and ##\frac{1}{2}(I-A)## are idempotent, satisfying the condition ##B^2 = B## for idempotents.
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You're given an involutory matrix A, which means that ##A^2 = I##. Clearly A can be decomposed into ##\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##.christang_1023 said:Homework Statement: A square matrix ##A## is said to be
• an involutory matrix if ##A^2 = I##,
• an idempotent if ##A^2 = A##.
Show that every involutory matrix can be expressed as a difference of two idempotents.
Homework Equations: The involutory matrix can be decomposed to ##A=\frac{1}{2}(I+A)-\frac{1}{2}(I-A)##
I feel confused about proving the two terms are idempotents.