Suppose A is a square matrix, and supoose isan eigenvalue for A with correspondi

Question

Suppose A is a square matrix, and supoose isan eigenvalue for A with corresponding eigenvectorX. (i) Show that 2 is an eigenvalue forA2 with corresponding eigenvector X. (ii) Suppose A2=I. Shoe that is either -1 or 1. (iii) Suppose A2=A. Show that is either 0 or 1. (iv) Suppose A2= 0. Show that=0 Suppose A is a square matrix, and supoose isan eigenvalue for A with corresponding eigenvectorX. (i) Show that 2 is an eigenvalue forA2 with corresponding eigenvector X. (ii) Suppose A2=I. Shoe that is either -1 or 1. (iii) Suppose A2=A. Show that is either 0 or 1. (iv) Suppose A2= 0. Show that=0

Explanation / Answer

[Note : The roots of the minimal polynomial are theeigenvalues.] These points will be used throughout, so it is important tounderstand these.
(i) A2x = A (Ax) = A (x)      = (Ax) = (x)      = 2x (ii) A2 = I implies A2 - I =0. So A satisfies x2 - 1 = 0. The minimal polydivides x2 - 1. So it is either x - 1 or x + 1 orx2 - 1. Hence the roots of the minimalpoly are ±1.
(iii) A2 = A implies A2 - A =0. So A satisfies x2 - x = 0.The minimal poly divides x2 - x. So it iseither x = 0 or x - 1= 0 or x2 - x. Hence theroots of the minimal poly are 0,1.
(iv) A2 = 0. So A satisfiesx2 = 0. The minimal poly dividesx2. So it is either x or x2. Hence theroots of the minimal poly is 0.

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