Question

Suppose A is a 2×2 matrix such that A^{2} =
I_{2} and let J be the Jordan form of A. Show that
J^{2} = I_{2} and use this fact to conclude that J
is diagonal

Answer #1

we have given that A^{2}=I where A is a 22
matrix and also I is the identity matrix of order 2

**as we know A ^{2}=I imply A is involutary
matrix**

**and a involutary matrix has eigen value 1 or -1 as its
determinant is equal to 1 or -1 by above condition**

**A ^{2}=I imply (A-I)(A+I)=0 imply either |A-I|=0
OR |A+I|=0**

hence its eigen values are 1 and -1 and its jordan form will be

J= where J represent the jordan form of A

and J^{2}==

**HENCE we get J ^{2}=I where J is the diagonal
matrix and I is the identity matrix of order 2**

A square matrix A is said to be idempotent if A2 = A. Let A be
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Show that I − A is also idempotent.
Show that if A is invertible, then A = I.
Show that the only possible eigenvalues of A are 0 and 1.(Hint:
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Show that TA(x) = projW x and TI−A(x)...

A square matrix A is said to be idempotent if
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Show that I − A is also
idempotent.
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associated eigenvalue λ and then multiply
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