Let $f:V\to \mathbb{R}$ be a quadratic function and $V$ is a vector space with $\dim V=n$. We say that $f$ is positive semi-definite if for all $x \in V$ we have $f(x)\geq 0$.

I know Sylvester's law which states the following: $f$ is positive definite iff all the following matrices have a positive determinant: the upper left 1-by-1 corner of $M$, the upper left 2-by-2 corner of $M$, the upper left 3-by-3 corner of $M$,..., $M$ itself, where $M$ is a matrix of $f$. In other words, all of the leading principal minors must be positive.

I need to show that $f$ is positive semi-definite iff all the principal minors (we have only $2^n-1$ of them) are non-negative.

I was not able to find the proof of this in MSE.

So I would be very grateful if someone can give detailed proof of this fact.


Here's an outline of the proof that a real square symmetric matrix is positive semi-definite iff every principal minor $\ge 0.$ Whenever I write matrices and/or row/column matrices as being multiplied, there is an implicit assumption that the sizes are appropriate for multiplication. $$*$$ Lemma1: If $M$ is a real square symmetric matrix with negative determinant then $$w^TMw<0 \text { for some column-vector } w \ne 0 $$

Proof: For some invertible matrix $P$, $$P^TMP= \text {diag}(d_1,...,d_n).$$ Taking determinants, we see that at least one $d_i$ must be negative. Thus $$v^TP^TMPv<0$$ for some $v \ne 0.$ Let $w=Pv$ Q.E.D. $$*$$ Lemma 2: If $A$ is a real $n \times n$ symmetric matrix and $v^TAv \ge 0$ for all column vectors, then every principal minor of $A$ is non-negative. $$*$$ Proof: Suppose $M$ is an $s \times s$ principal sub-matrix of $A $ where $1 \le s \le n$ such that $\det M<0.$ Let $w \ne 0$ be such that $w^TMw<0 $ Let $w’$ be a column-vector of size $n$ obtained by by using the same entries as those in $w$ for indices that occur for $M$ and putting all other entries =0. Then $$w’^TAw’=w^TMw<0$$, a contradiction. Q.E.D. $$*$$ Lemma 3: If $A$ is a real $n \times n$ symmetric matrix and every principal minor of $A$ is non-negative, then $A$ is positive semi-definite. $$*$$ Proof: We assert that if $t>0$ then $tI+A$ is positive-definite. Consider the determinant of the upper left $s \times s$ corner of $tI+A$ where $1 \le s \le n$. It is $\det(tI+M)$ where $M $ is the upper left $s \times s $ corner of $A$. Note that every principal minor of $M$ is a principal minor of $A$. Then $$\det(tI+M)=t^s+\sum_{i=1}^s{}(\sum \text {principal minors of order $i$ of $M$ })t^{s-i}>0 $$ which proves our assertion. Suppose $$v^TAv=-c \text { where }c>0.$$ Then $v \ne 0 $ Let $$\xi=\frac{c}{v \bullet v} $$ which is the same as $$\xi v^TIv=c$$ Thus $$v^TAv=-\xi v^TIv$$ $$v^T(\xi I+A)v=0$$, a contradiction. Q.E.D.

  • $\begingroup$ Regarding Lemma 1 since some $d_i$ is negative then I can take $v$ to be the vector with $i$th coordinate 1 and other are zero, right? $\endgroup$ – ZFR Mar 5 '20 at 2:53
  • $\begingroup$ @ZFR:Yes, that is correct. $\endgroup$ – P. Lawrence Mar 5 '20 at 3:01
  • $\begingroup$ Let me ask you one more question: could you clarify the formula for $\det(tI+M)$? I didn't get how you derived the RHS of it? Seems unclear to me. Thanks! $\endgroup$ – ZFR Mar 5 '20 at 13:47
  • $\begingroup$ I would be very grateful if you can give more details about RHS of $\det(tI+M)$, please. $\endgroup$ – ZFR Mar 5 '20 at 16:14
  • 1
    $\begingroup$ @ZFR: I wish I could say I came up with the proof by personal brilliance but in fact I took it from ALGEBRA A Text-Book of Determinants, Matrices, and Algebraic Forms by W. L. Ferrar, an old book that has aged very well. I treasure my copy, which was the first mathematics book I ever bought, other than a course text, in 1961. $\endgroup$ – P. Lawrence Mar 7 '20 at 20:30

Claim: If all principal minors of $A$ are non-negative then $A$ is positive semidefinite.

Proof: Let $S_k$ denote the sum of all the k principal minors of $A$ and let $\lambda_1,\dots,\lambda_n$ denote the eigenvalues of $A$ which are real as $A$ is symmetric.

Define the polynomial $p$ as $p(t)= \det(tI + A)$.

$p$ is monic and $(-1)^np(-t)$ is the characteristic polynomial of $A$ from which it follows $p(t) = (t+ \lambda_1)\dots(t+\lambda_n).$

Let $e_i$ denote the vector with $1$ at the ith position and zero elsewhere. Let $a_i$ denote the ith column of $A$, then

$$tI + A = \begin{bmatrix} te_1 + a_1 & te_2 + a_2 & \dots & te_n + a_n \end{bmatrix}.$$

By multilinearity of the determinant $p(t) = \sum_{r=0}^n t^r S_{n-r}$ where $S_k$ denotes the sum of all principal minors of $A$ of order $k$.

Suppose all principal minors are non-negative, then $S_k \geq 0$ for all $k$.

Consequently the polynomial $$ p(x) = (x+\lambda_1)(x+\lambda_2)\dots (x+\lambda_n) = x^n + S_{1}x^{n-1} + \dots + S_{n-1}x + S_n $$ has only non-negative coefficients. This means $p(x) > 0$ for $x >0$ so no real root of $p(x)$ can be larger than $0$. But all the roots of $p(x)$ are real, and equal the $-\lambda_i$'s, so we must have$ -\lambda_i \leq 0$ or $\lambda_i \geq 0$ for all $i$. (This is from here.) Hence $A$ is positive semi-definite as all its eigenvalues are zero or larger.

Conversely let $A$ be positive semi-definite. And let $B$ be any principal submatrix of $A$, then $B$ is positive semi-definite as $x^TBx \geq 0$ for all $x$, and hence its eigenvalues are non-negative, and hence $\det(B) \geq 0.$

  • $\begingroup$ You know there is one moment in your proof which seems very unclear to me. By $S_r$ you denoted the sum of all principal minors of the size $r$. Then somehow you got the different expression for $S_r$ through eigenvalues. You really need to clarify this moment. It is not so obvious. $\endgroup$ – ZFR Mar 25 '20 at 19:42
  • $\begingroup$ @ZFR I updated the proof. $\endgroup$ – Arin Chaudhuri Mar 30 '20 at 19:11

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