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I have to show that all $2\times2$ sub matrices

$\left[ \begin{array}{rr} X_{ii} & X_{ij} \\ X_{ij} & X_{jj} \\ \end{array}\right]$

of a real, positive definite symmetric matrix $X$ are positive semi definite. After that I would like to proof that for this $2\times 2$ sub matrix the inequality $X_{ii}X_{jj}-X_{ij}^{2}≥0$ holds.

Can anybody help me?

I've tried to do so with the definition of positive and positive semi definite matrices (positive definite iff $x^{T}Ax>0$ $\forall x\neq 0 $), but I am desperate.

Thanks for your help in advance.

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    $\begingroup$ I don't see how that can be true. You can start with a $4\times 4$ identity matrix, and add any $2\times 2$ sub-matrices $A$ and $A^T$ in the top right and bottom left corners; and by making $A$ small enough, you can retain the positive definiteness of the $4\times 4$ matrix. Did you perhaps forget to say that $X$ is a $3\times 3$ matrix? Or am I missing something? $\endgroup$
    – TonyK
    Jul 24, 2022 at 11:48
  • $\begingroup$ Thanks for your comment :) No, I definitely have to proof this for $n\times n$ matrices which are per definition positive definite and symmetric. Maybe I should add that I have to proof this for all $2\times 2$ sub matrices which have diagonal elements in it. $\endgroup$
    – lolikoli
    Jul 24, 2022 at 11:56
  • $\begingroup$ OK, it makes sense now. $\endgroup$
    – TonyK
    Jul 24, 2022 at 15:15

1 Answer 1

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Can you clarify what you mean by "sub-matrices" more precisely? Perhaps you mean $2 \times 2$ principal minors (i.e. submatrices where we select the same two rows and columns).

Hint 1. For simplicity, suppose that $A$ is the top left $2 \times 2$ principal minor of $X$: consider $z = (z_1, z_2, 0, \dots, 0)$, and observe that $$ z^T X z = (z_1, z_2)^T A (z_1, z_2) . $$ What can you say about this? Generalise this to arbitrary $2 \times 2$ (principal minor) submatrices in the obvious way.

Hint 2. Suppose we know that all $2 \times 2$ (principal minor) submatrices are positive semi-definite. Let $A$ be any such submatrix, say $$ A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{pmatrix} . $$ Observe that $\det(A) = a_{11} a_{22} - a_{12} a_{21} = a_{11} a_{22} - a_{12}^2$ if $A$ is symmetric (this is another reason why principal minor makes sense). This looks like the expression you want. What do you know about the relationship between the determinant of a matrix, its eigenvalues, and how does this relate to positive semi-definiteness?

P.S. You may want to look up Sylvester's criterion.

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  • $\begingroup$ Hi, I've clarified what I mean by sub-matrices. Regarding your hint 1: $z^{T}Xz=(z_1, z_2)^{T}A(z_1,z_2)$ is definitely positive as X is positive definite. Regarding your hint 2: A matrix is positive semi definite if all the eigenvalues are equal or bigger than zero. The determinant is the product of all its eigenvalues. Ah, I think I've got it. $\endgroup$
    – lolikoli
    Jul 24, 2022 at 12:08
  • $\begingroup$ That is indeed a $2 \times 2$ principal minor :) $\endgroup$
    – JKL
    Jul 24, 2022 at 12:09
  • $\begingroup$ One question left regarding your first hint: As $z^{T}Xz=(z_1,z_2)^{T}A(z_1,z_2)$ is positive, we can conclude that all $2\times 2$ sub-matrices are positive definite, can't we? But my book states they are positive semi-definite. One question regarding your second hint: Suppose that the matrix is positive semi-definite, then all eigenvalues are greater or equal than zero and as the determinant is the product of the eigenvalues, we also know that the determinant is greater or equal than zero, isn't it? Then I get the expression I want. $\endgroup$
    – lolikoli
    Jul 24, 2022 at 12:35
  • $\begingroup$ 1. Positive definite matrices are positive semi-definite. It seems that this stronger result is possible then. 2. That sounds right. $\endgroup$
    – JKL
    Jul 24, 2022 at 12:58

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