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Questions tagged [positive-semidefinite]

Relating to a symmetric $n\times n $ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$

34
votes
3answers
48k views

Is the product of symmetric positive semidefinite matrices positive definite?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? ...
16
votes
3answers
30k views

Prove that every positive semidefinite matrix has nonnegative eigenvalues

There is a theorem which states that every positive semidefinite matrix only has eigenvalues $\ge0$ How can I prove this theorem?
3
votes
1answer
898 views

How to prove that $A$ is positive semi-definite if all principal minors are non-negative?

Let $A\in\mathbb C^{n\times n}$ be a Hermitian matrix such that all its principal minors are non-negative (i.e. for $B=\left(a_{l_il_j}\right)_{1≤i,j≤k}$ with $1≤l_1<...<l_k≤n$ we have $\det(B)≥...
48
votes
5answers
12k views

Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is ...
2
votes
1answer
196 views

How to prove positive semidefiniteness of two matrices through Schur Complement?

Let matrices $X$ and $Y$ be positive semidefinite. Show that $X \succeq Y \succ 0$ is equivalent to $Y^{-1} \succeq X^{-1}$. The teacher tells me that it is easy to prove via the Schur complement. But ...
8
votes
1answer
6k views

$f$ is convex function iff Hessian matrix is nonnegative-definite.

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, $f \in C^2$. Show that $f$ is convex function iff Hessian matrix is nonnegative-definite. $f(x,y)$ is convex if $f( \lambda x + (1-\lambda )y) \le \...
4
votes
1answer
193 views

Definiteness of a general partitioned matrix $\mathbf M=\left[\begin{matrix}\bf A & \bf B\\\bf B^\top & \bf D \\\end{matrix}\right]$

If $\mathbf M=\left[\begin{matrix}\bf A & \bf b\\\bf b^\top & \bf d \\\end{matrix}\right]$ such that $\bf A$ is positive definite, under what conditions is $\bf M$ positive definite, positive ...
7
votes
1answer
1k views

Positive semidefinite cone is generated by all rank-$1$ matrices.

The positive semidefinite cone is generated by all rank-$1$ matrices $xx^T$, which form the extreme rays of the cone. Positive definite matrices lie in the interior of the cone. Positive semidefinite ...
5
votes
1answer
3k views

The product of two symmetric, positive semidefinite matrices has non-negative eigenvalues

How can I prove the following? If $A$ and $B$ are two symmetric, positive semidefinite matrices then all eigenvalues of $AB$ are non-negative.
7
votes
2answers
16k views

Checking if a matrix is positive semidefinite

Determine whether the following $2 \times 2$ matrix is positive semidefinite (PSD) $$\begin{bmatrix}\frac{2}{x} & \frac{-2y}{x^2} \\\frac{-2y}{x^2} & \frac{2y^2}{x^3}\end{bmatrix}$$ ...
17
votes
3answers
1k views

Are positive definite matrices robust to “small changes”?

Let $A$ be a positive-definite matrix and let $B$ be some other symmetric matrix. Consider the matrix $$ C=A+\varepsilon B. $$ for some $\varepsilon>0$. Is it true that for $\varepsilon$ small ...
6
votes
1answer
5k views

Decomposition of a positive semidefinite matrix

Let $Y \in \mathbb{R}^{n \times n}$ be a symmetric, positive semidefinite matrix such that $Y_{kk} = 1$ for all $k$. This matrix is supposed to be factorized as $Y = V^T V$, where $V \in \mathbb{R}^{...
3
votes
1answer
489 views

How to write the dual of quadratic program with positive semidefinite matrix?

Consider the following quadratic program $$\begin{align*}&\text{min } \frac{1}{2}b^TDb+d^Tb\\ &\text{s.t. } Ab\le b_0\end{align*}$$ where $D$ is a positive semidefinite matrix. The ...
2
votes
1answer
878 views

Hessian matrix for convexity of multidimensional function

To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the ...
5
votes
3answers
567 views

Does the Schur complement preserve the partial order?

Let $$\begin{bmatrix} A_{1} &B_1 \\ B_1' &C_1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} A_2 &B_2 \\ B_2' &C_2 \end{bmatrix}$$ be symmetric positive definite and ...
4
votes
1answer
258 views

Variables that will make this matrix positive semi-definite

I have a matrix $$M=\begin{bmatrix} 1+t+m &n&t+n&m+c \\ n &1+t-m&m-c & t-n \\ t+n & m-c&1-t-m & -n \\ m+c & t-n & -n & 1-t+m \end{bmatrix}$$ where I ...
3
votes
2answers
78 views

Local minimum has neighbourhood with positive semi-definite Hessian?

I know that any local minimum $x_0$ of a (twice continuously differentiable) function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. Can we say ...
2
votes
1answer
68 views

Stability of a matrix product

Let $H$ be a real, invertible and positive semi-definite matrix, in the sense that its symmetric part $S$ is positive semi-definite. Consider the matrix $$ G = (I+\alpha H_d)H $$ for some $\alpha > ...
2
votes
1answer
198 views

A question on positive semi definite matrices

Suppose $A,B$ are symmetric, positive semi-definite matrices of same order such that $A \preceq B \preceq \kappa A$. How to prove that this is equivalent to $\frac{1}{\kappa}A^+ \preceq B^+ \preceq A^+...
1
vote
1answer
430 views

Find the Matrix Projection of a Symmetric Matrix onto the set of Symmetric Positive Semi Definite (PSD) Matrices

Consider $\mathbb{S}^n$, the set of all $n\times n$ real symmetric matrices. Let $A\in \mathbb{S}^n$ and $A=U\Lambda U^{\top}$ be its spectral decomposition. I want to know how to prove that $\Pi_{\...
3
votes
0answers
415 views

Constrained quadratic programming with positive semidefinite matrix

I am facing the following problem: $$\begin{equation*} \begin{aligned} & \underset{z}{\text{minimize}} & & \textbf{z}^T \textbf{Q} \textbf{z} + \textbf{b}^T \textbf{z}\\ & \text{...
3
votes
2answers
3k views

Ensuring that a symmetric matrix with nonnegative elements is positive semidefinite

I have the following matrix $A$: symmetric all positive and/or zero values the main diagonal is all the same value, $x$. To ensure that the matrix $A$, is positive semidefinite, must I only ensure ...
2
votes
1answer
74 views

Completion of $2 \times 2$ positive semidefinite rank-$1$ partial matrix

This question is related to one property of rank-$1$, positive semidefinite matrices. Would be very useful in SDP problems (which is where I found it). Consider a $3 \times 3$ positive semidefinite ...
1
vote
1answer
37 views

How to show $\text{Tr}(AB) \leq \text{Tr}(AC)$ where $B \preceq C$?

Given three positive semi-definite matrices $A, B, C$. Show $\operatorname{Tr}(AB) \leq \operatorname{Tr}(AC)$ where $B \preceq C$? This inequality is the matrix form of multiplying a positive ...
1
vote
1answer
87 views

Parameters that will make this matrix positive semidefinite

I have a matrix $$M=\begin{bmatrix} 1+t+m &n&t+n&m+c \\ n &1+t-m&m-c & t-n \\ t+n & m-c&1-t-m & -n \\ m+c & t-n & -n & 1-t+m \end{bmatrix}$$ where I ...
1
vote
1answer
558 views

Are these positive semidefinite matrices invertible?

Suppose $n \ge 2$ and we have an $n \times n$ Hermitian, positive semidefinite matrix with $1$ down the main diagonal, and all off-diagonal entries have modulus strictly less than $\frac{1}{n-1}$. Can ...
1
vote
1answer
274 views

Eigenvalues of product of two symmetric positive semi-definite real matrices.

I know that if $A$ and $B$ either A and B are positive definite matrices then the product matrix $AB$ is similar to $A^{\frac{1}{2}}BA^{\frac{1}{2}}$ and those two matrices have same eigenvalues. This ...
1
vote
1answer
53 views

Does $ 0 \leq A \leq B \implies 0\leq B^\dagger \leq A^\dagger$?

The question is stated in the title. Where $A$ and $B$ are positive definite real matrices and $\dagger$ stands for the Moore Penrose inverse. This question link1 seems to be close but it has an ...
0
votes
1answer
53 views

Local minimum of an analytic function

This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. ...
0
votes
1answer
46 views

For a matrix, how to prove its max eigenvalue max inner product with any matrix which is semidefinite and has a trace less than 1?

How to prove the equation in (22) in 1801.09344? For simplicity, I re-describe it here: $$\max_{P \succeq0, \operatorname{trace}(P) \leq 1}\! \langle A,P\mkern1.5mu\rangle= \lambda_+(A)$$ where $\...
0
votes
1answer
474 views

Positive definite sequence and its corresponding determinant.

I am currently reading the book of Akhiezer http://www.maths.ed.ac.uk/~aar/papers/akhiezer.pdf and I saw on page 1 that a sequence $\{s_n\}_{n=0}^\infty$ of real numbers is positive definite if $$ \...