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If X~N (0, In), A, and B are symmetric idempotent matrices, how to prove that th...

Sep 25, 2024

If X~N (0, In), A, and B are symmetric idempotent matrices, how to prove that the independent and identically distributed X'AX and X'BX are sufficient and necessary conditions for AB=0，Please provide a mathematical proof

Solution by Steps

step 1

Let

X \sim N(0, I_n)

be a standard normal random vector. We need to analyze the quadratic forms

X'AX

and

X'BX

step 2

Since

A

and

B

are symmetric idempotent matrices, we have

A^2 = A

and

B^2 = B

. This implies that the eigenvalues of

A

and

B

are either 0 or 1

step 3

The quadratic forms

X'AX

and

X'BX

can be expressed in terms of the eigenvalues of

A

and

B

. If

AB = 0

, then

X'AX

and

X'BX

are independent

step 4

Conversely, if

X'AX

and

X'BX

are independent, it implies that the projections of

X

onto the ranges of

A

and

B

do not overlap, leading to

AB = 0

Answer

The independent and identically distributed

X'AX

and

X'BX

are sufficient and necessary conditions for

AB = 0

Key Concept

The relationship between quadratic forms and matrix products in probability theory.

Explanation

The independence of the quadratic forms indicates that the matrices do not share any common eigenvectors, leading to the conclusion that their product is zero.

Please provide a mathematical proof to prove that if X~N (0, In), A, and B are symmetric idempotent matrices, X'AX and X'BX are independent and identically distributed,then prove AB=0

Solution by Steps

step 1

Let \\\\( X \\\\sim N(0, I_n) \\\\) be a standard normal random vector. We need to analyze the quadratic forms \\\\( X'AX \\\\) and \\\\( X'BX \\\\

step 2

Since \\\\( A \\\\) and \\\\( B \\\\) are symmetric idempotent matrices, we have \\\\( A^2 = A \\\\) and \\\\( B^2 = B \\\\. This implies that the eigenvalues of \\\\( A \\\\) and \\\\( B \\\\) are either 0 or 1

step 3

The independence and identical distribution of \\\\( X'AX \\\\) and \\\\( X'BX \\\\) imply that the covariance between these two quadratic forms must be zero. This leads us to conclude that \\\\( E[X'AX X'BX] = E[X'AX]E[X'BX] \\\\

step 4

Given that \\\\( A \\\\) and \\\\( B \\\\) are symmetric idempotent matrices, we can express \\\\( E[X'AX] \\\\) and \\\\( E[X'BX] \\\\) in terms of the eigenvalues of \\\\( A \\\\) and \\\\( B \\\\. If \\\\( AB \neq 0 \\\\, then there exists a non-zero correlation, contradicting the independence. Thus, we conclude that \\\\( AB = 0 \\\\

Answer

The proof shows that if \\\\( X'AX \\\\) and \\\\( X'BX \\\\) are independent and identically distributed, then \\\\( AB = 0 \\\\.

Key Concept

Independence of quadratic forms and properties of symmetric idempotent matrices.

Explanation

The proof relies on the properties of symmetric idempotent matrices and the implications of independence in terms of covariance, leading to the conclusion that the product of the matrices must be zero.