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Questions tagged [positive-semidefinite]

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

49
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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 ...
35
votes
3answers
49k 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? ...
30
votes
2answers
48k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m]...
18
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 ...
18
votes
3answers
31k 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?
18
votes
1answer
5k views

Why do mathematicians use only symmetric matrices when they want positive semi-definite matrices?

Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices? I mean I haven't seen using non-symmetric positive semi/definite matrices. If non-symmetric positive ...
11
votes
3answers
15k views

Does a positive semidefinite matrix always have a non-negative trace?

A simple question: If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace. If not, any counter-example? ...
11
votes
1answer
1k views

Why is this determinant positive?

I have seen that the $k$-dimensional volume of an parallelepiped in $\mathbb{R}^n$, i.e., $$P(v_1, \ldots, v_k) = \{t_1v_1 + \dotsb + t_kv_k : 0 \le t_i \le 1 \}$$ is $\sqrt{\det(T^{\top}T)}$, where $...
10
votes
1answer
117 views

Meaning of $x^T A x$

I've seen the term $x^T A x$ come up in a bunch of different areas of linear algebra, where A is a square and usually symmetric matrix. Places I've seen it include defining the Raleigh quotient, ...
9
votes
3answers
338 views

Determining if a symmetric matrix is positive definite

I have a symmetric matrix where all non-diagonal elements are positive and identical, and all diagonal elements are identical as well. For example, the $3 \times 3$ version of this matrix has the ...
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 \...
8
votes
1answer
97 views

Probability that a random binary matrix is positive semi-definite

Let $A$ be a random $n \times n$ matrix such that $A_{ij}\in\{0,1\}$. Assume that each element $A_{ij}$ equals 1 with some probability $p>0$ and that all the draws are independent across elements. ...
7
votes
2answers
296 views

Difficulty in linear algebra question.

Let $a_{ij}=a_ia_j$ $1\leq i,j\leq n$ where $a_1,... a_n$ are real numbers. Let $A=(a_{ij})$ be the $n \times n $ matrix . Then It is spoosible to choose $a_1,... a_n$ so as to mke A non singular. ...
7
votes
3answers
3k views

Is $U=V$ in the SVD of a symmetric positive semidefinite matrix?

Consider the SVD of matrix $A$: $$A = U \Sigma V^\top$$ If $A$ is a symmetric, positive semidefinite real matrix, is there a guarantee that $U = V$? Second question (out of curiosity): what is the ...
7
votes
2answers
17k views

Checking if a matrix is positive semidefinite

Determine whether the following $2 \times 2$ matrix is positive semidefinite (PSD) $$\begin{bmatrix}\frac{2}{x} & \frac{-2y}{x^2} \\\frac{-2y}{x^2} & \frac{2y^2}{x^3}\end{bmatrix}$$ ...
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 ...
7
votes
1answer
73 views

If a matrix $Q$ is symmetric and positie definite, is it possible to show that the matrix $Q-A^T(AQ^{-1}A^T)^{-1}A$ is also positive definite?

If I have a symmetric and positive definite $n\times n$ matrix $Q$ and a full row-rank totally unimodular $m\times n$, where $m<n$, matrix $A$, is it posible to show that the matrix $$Q-A^T(AQ^{-...
7
votes
2answers
2k views

Criterion for positive semidefinite matrices

Is there a criterion for positive semidefiniteness of a matrix in terms of dimension reduction, i.e, such that positive semi-definiteness of $n \times n$ matrix is expressed as positive ...
6
votes
1answer
6k 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}^{...
6
votes
3answers
616 views

Is a sinc-distance matrix positive semidefinite?

I've been trying to crack this problem for days but I can't find a way around it. Given a set of unique N points $X = {x_1,..x,N}, x_i \in R^3$, the associated sinc-distance matrix $S \in R^{n\times n}...
6
votes
1answer
124 views

Counter-example for a matrix not being a correlation matrix

Is there an example of an $n \times n$ matrix that: is real-valued and symmetric with entries between $-1$ and $+1$ has diagonal elements equal to $1$ has a non-negative determinant has non-negative ...
6
votes
1answer
133 views

Trace inequality for a product of p.s.d. matrices and their pseudo inverse.

Let $A, B_i$ be positive semidefinite real matrices. Let $\dagger$ stand for the Moore-Penrose generalized inverse. I managed to prove that if $\operatorname{Ran}B_1\subseteq\operatorname{Ker}B_2$ ...
6
votes
1answer
122 views

Characterizing duals of cones that are linear images of the positive semidefinite cone

Let $M_n$ denote the space of $n\times n$ matrices over complex numbers. The space of self-adjoint matrices is denoted $$ M_n^{sa} = \{A\in M_n\, :\, A^*=A \}, $$ where $A^*$ denotes the conjugate ...
6
votes
1answer
125 views

Existence of a positive semidefinite matrix that satisfies a set of equality constraints

Given vectors $a_1, b_2, a_2, b_2 \in \mathcal{R}^{n\times 1}$, I am interested in finding a positive semi-definite matrix $M \in \mathcal{R}^{n\times n}$, $M \succeq 0$, such that $M\cdot a_1 = b_1$, ...
5
votes
3answers
374 views

If a matrix $A$, not necessarily symmetric, has real, nonnegative eigenvalues, is it positive semidefinite?

We know that a symmetric matrix $A$ is positive semidefinite i.e. $x^TAx \geq 0$ if and only if all its eigenvalues are nonnegative. Now suppose I have a matrix (not necessarily symmetric) $A$, ...
5
votes
3answers
582 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 ...
5
votes
1answer
2k views

Is the outer product of a column vector with itself positive semi definite? [closed]

Say we have a column vector $x=[x_1\ x_2\ x_3]^T$. Then is $ xx^T $ positive semi definite.
5
votes
2answers
334 views

Minimize $\mbox{trace}(AX)$ over $X$ with a positive semidefinite $X$

I want to minimize $\mbox{trace}(AX)$ over $X$, under the constraint that $X$ is positive semidefinite. I guess the solution should be bounded only for a positive semidefinite $A$, and it's zero, or ...
5
votes
2answers
45 views

$k\sum v_i v_i^T-\big(\sum v_i\big)\big(\sum v_i^T\big)\succeq 0$

My professor claimed that $$k\sum_{i=1}^k v_i v_i^T-\Big(\sum_{i=1}^k v_i\Big)\Big(\sum_{i=1}^k v_i^T\Big)\succeq 0,$$ holds for any family of vectors $\{v_1,\dots,v_k\}$, and can be shown using the ...
5
votes
1answer
176 views

Trace inequality on the product of positive semi-definite matrices

Let $A_1$ and $A_2$ be positive semi-definite matrices such that Tr$(A_1) \leq$ Tr$(A_2)$. Let $B$ be another positive semi-definite matrix. Is it true that Tr$(A_1B) \leq$ Tr$(A_2B)$?
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.
5
votes
2answers
92 views

What is the Hessian of the spectral norm?

The spectral norm of a symmetric matrix is the absolute value of the top eigenvalue. The gradient of this norm is $uu^T$ where $u$ is the eigenvector associated with that top eigenvalue. Assume that $...
5
votes
1answer
73 views

What is the formula for projection onto spectraplex?

A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let $$ S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
5
votes
1answer
95 views

Required conditions for eigenvalues $\lambda_{\min}(A) >\lambda_{\min}(B)$ and $\lambda_{\max}(A) >\lambda_{\max}(B) $ etc?

Given $\operatorname{tr}(X^TAX) > \operatorname{tr}(X^TBX)$ and $A$ and $B$ are p.s.d then under what conditions will we have $\lambda_{\max}(A)>\lambda_{\max}(B)$ to be guaranteed ? What are ...
5
votes
1answer
295 views

Matrix Inverse is a Uniformly Continuous function for Uniformly Positive Definite matrices?

Definition 1: A collection of $k\times k$ positive semidefinite matrices $\{A_n\}$ is said to be uniformly positive definite if for some $\eta > 0$, $det(A_n) > \eta$. Definition 2: A function $...
5
votes
1answer
124 views

Matrices whose all principal $k\times k$ sub-matrices are positive semidefinite

I would like to know whether the set of $n\times n$ Hermitian matrices whose all ${{n}\choose{k}}$ principal $k\times k$ sub-matrices---the matrices obtained by removing $n-k$ columns as well as the ...
4
votes
1answer
779 views

When is $\det(A+B)=\det(A)+\det(B)$ for positive definite $A$ and positive semidefinite $B$?

Let $A$ be a positive definite matrix and $B$ be a positive semi-definite matrix. Under what conditions does $\det(A+B)=\det(A)+\det(B)$ hold? We all know that for any two positive semi-definite ...
4
votes
1answer
259 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 ...
4
votes
2answers
2k views

Properties of symmetric projection matrices

A square matrix $P$ is called a symmetric projection matrix if $P = P^T$ and $P ^2 = P$. Show that a symmetric projection matrix $P$ satisfies the following properties. $\|x\|^2=\|Px\|^2+\|(1-...
4
votes
1answer
2k views

Prove that $A\ge0, B\ge0$ and $A\ge B$ implies $B^{-1}\ge A^{-1}$

Does anyone know how to prove the following: Suppose $A$ and $B$ are both positive definite and $A - B$ is positive semi-definite. Show that $B^{-1} - A^{-1}$ is also positive semi-definite. ...
4
votes
1answer
42 views

$f(t)=\text{det}(A+tB)$ is continuous for symmetric positive semi-definite matrices $A$ and $B$?

We have any symmetric positive semi-definite matrices $A$ and $B$ where $\text{det}B=0$. A function $f(t)=\text{det}(A+tB)$ is continuous on $t\in[0,1]$? $f(t_{1})-f(t_{2})=\text{det}(A+t_{1}B)-\...
4
votes
1answer
211 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 ...
4
votes
2answers
780 views

When does a real, positive definite matrix have positive entries?

Let $A = (a_{ij})_{i,j=1,\dots,n}$ be a real symmetric square matrix. Suppose $A$ is positive definite. Are there sufficient conditions that guarantee $a_{ij} > 0$ for all $i,j = 1,\dots, n$? I ...
4
votes
1answer
179 views

Differentiability of the Schatten $p$-norm on positive definite matrices

Let $V$ be the vector space of symmetric matrices in $\Bbb R^{n\times n}$. For $p\in (1,\infty)$, the Schatten $p$-norm of $M\in V$ is defined as $\|M\|_p =(\sum_{i=1}^n \sigma_i(M)^p)^{1/p}$ where $\...
4
votes
1answer
80 views

What is the apex angle of the cone of positive semidefinite matrices?

Let $\def\S{\mathbf S}\S^n$ be the linear space of symmetric $n \times n$ matrices and $\S_+^n$ be the subset of positive semidefinite matrices. It is well-known that $\S_+^n$ is a convex cone in $\S^...
4
votes
1answer
63 views

Matrices Inequality Proof

Recently, I read a paper and there is a step which turns out not obvious to me. The statement is as follows: All matrices here are real matrices. $F$ is an arbitrary square matrix. $\Psi$ is a ...
4
votes
0answers
49 views

Show that the matrix $\big(\frac{1}{a_i + a_j}\big)_{n\times n}$ is positive semidefinite [duplicate]

Given positive real numbers $a_1,\dots ,a_n$, how can I prove that the symmetric matrix composed of the entries $\frac{1}{a_i + a_j}$ is positive semi-definite *this is not a hw question but a self ...
4
votes
0answers
4k views

Checking positive semidefiniteness in MATLAB

Let $\mathbf{A}$ be a $n\times n$ matrix. I want to check in MATLAB if it is PSD or not. Which tests, in MATLAB, should I do for this purpose? I know that if $\mathbf{A}$ is PSD then following holds ...
3
votes
5answers
375 views

Show that if $A$ is positive definite then $A + A^{-1} - 2I$ is positive semidefinite

Let $A$ be a real symmetric positive definite matrix. Show that $$A + A^{-1} -2I$$ is positive semidefinite. I found that $A^{-1}$ is a positive definite matrix, thus $A + A^{-1}$ is also a positive ...
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 ...