Question

Matrix problem

Answer 1

If \[A \in \mathbb{R} ^{n x m}\], and \[\exists(A ^{T}*A)^{-1}\] , which of these statements are true? Justify your answer : \[a) (A ^{T}*A)^{-1}\] is a symmetric matrix; \[b) (A ^{T}*A)^{-1} \] is an anti-symmetric matrix ; \[c) P ^{2} = P\]; \[d)P\] is symmetric ; \[e) P \in \mathbb{R} ^{nxn}\]

Answer 2

about a) and b): I'd say a) is right.. First: Think of, what it means to be symmetrical. if a matrix A is symmetrical then:\[A ^{T}=A.\]
So we have to check: \[(A ^{T}A)^{-1}=((A ^{T}A)^{-1})^{T}.\]\[((A ^{T}A)^{-1})^{T}=((A ^{T}A)^{T})^{-1}=(A ^{T}(A ^{T}) ^{T})^{-1}=(A ^{T}A)^{-1}\] So here we see, a) is true, therefore b) is wrong.

about c,d,e: What is P?

Answer 3

SORRY :
\[P = A*(A ^{T}*A)^{-1}*A ^{T} \]

Which statements are true? Justify

c) \[P ^{2 } = P\]

d) P is a symmetric matrix

e) \[P \in \mathbb{R} ^{NXN}\]