[Postgraduate Mathematics] Proof and derivation: Let A and B be positive definite matrices of order m and n, respectively, then the block matrix C=[A, O, O, B] is a positive definite matrix

Source of the question: Tang Jiafeng 1800 Quadratic Type Question 8

Answer:

The conclusion is used here: if A and B are positive definite matrices of order m and n, respectively, then the block matrix $C=\left(\begin{array}{cc}A& O\\ & B\end{array}\right)$
is a positive definite matrix.

I first understood this with eigenvalues, but later I felt it was not very clear, so I used the definition to prove it, as shown in the figure below:

I referred to a paper published by Zhao Chenxia and others, and the author of the article gave a high judgment Order symmetric matrices are not a more concise method of positive definite matrices.


If you are interested, you can read this article

• Reference paper
  [1] Zhao Chenxia, ​​Cui Yuhuan, Chen Weili. Positive Definite Discrimination Method for a Class of Block Matrix [J]. Mathematical Learning and Research, 2010(5):1.