The problem asks to evaluate the correctness of the given statements, with the matrices $A$ and $B$ assumed to be positive definite $n \times n$ matrices.

Let's go through the statements one by one:

(A) $AB$ 的特征值都大于零 (The eigenvalues of $AB$ are all greater than zero)

For positive definite matrices $A$ and $B$, it's known that their eigenvalues are positive. However, the product $AB$ may not necessarily have all positive eigenvalues. Although $A$ and $B$ are individually positive definite, the product $AB$ is not necessarily positive definite (and may not even be symmetric). Therefore, we cannot conclude that all eigenvalues of $AB$ are positive.

Conclusion: False.

(B) $A = B$ 的充要条件是 $A^{-1}B$ 的特征值都是 -1 (The necessary and sufficient condition for $A = B$ is that all eigenvalues of $A^{-1}B$ are -1)

If $A = B$, then $A^{-1}B = I$, where $I$ is the identity matrix, whose eigenvalues are all 1, not -1. This statement seems to confuse the conditions for equality between $A$ and $B$.

Conclusion: False.

(C) $A^{-1} + B'$ 正定 ( $A^{-1} + B^T$ is positive definite)

Since $A$ and $B$ are both positive definite, their inverses and transposes are also positive definite. The sum of two positive definite matrices is also positive definite. Therefore, $A^{-1} + B'$ (which is $A^{-1} + B^T$) is positive definite.

Conclusion: True.

(D) $AB$ 正定 ( $AB$ is positive definite)

For positive definite matrices $A$ and $B$, their product is not necessarily positive definite unless the matrices commute (i.e., $AB = BA$). Since the problem does not specify that $A$ and $B$ commute, we cannot conclude that $AB$ is positive definite in general.

Conclusion: False.

Summary of Answers:

• (A) False
• (B) False
• (C) True
• (D) False

Would you like a detailed explanation on any specific part?

Here are 5 related questions:

1. What are the properties of positive definite matrices?
2. How do the eigenvalues of a matrix product relate to the eigenvalues of the individual matrices?
3. Under what conditions is the sum of two positive definite matrices also positive definite?
4. Can you give an example where the product of two positive definite matrices is not positive definite?
5. Why does the transpose of a positive definite matrix remain positive definite?

Tip: When dealing with eigenvalues of matrix products, always verify if the matrices commute before concluding any properties about the product.