Articles of positive definite

Prove that a non-positive definite matrix has real and positive eigenvalues

I have a $2\times2$ matrix $J$ of rank $2$ and a $2\times2$ diagonal positive definite matrix $Α$. Denote by $J^+$ a pseudoinverse of $J$. I can find many counterexamples for which $J^+AJ$ is not positive definite (e.g. $J=\left(\begin{smallmatrix}2&1\\1&1\end{smallmatrix}\right)$ and $A=\left(\begin{smallmatrix}1&0\\0&2\end{smallmatrix}\right)$), but for all of them $J^+AJ$ has real and positive eigenvalues. So, I was wondering […]

Maximize $\gamma$ such that $A+\gamma B\succeq 0$

Let $A \succ 0$ and $B$ be $n\times n$ symmetric matrices. Find the largest scalar $\gamma \geq 0$ such that $$A + \gamma B \succeq 0$$ Here is my (failed) attempt: The problem is: $$\max_{\gamma\ge 0} \gamma : v^\top(A+\gamma B)v \ge 0 \quad \forall v\in\mathbb R^n$$ Let $Au_i = \lambda u_i$ for $i=1,\dots,n$. And let […]

Is the matrix square root uniformly continuous?

Let $\operatorname{Psym}_n$ be the cone of symmetric positive-definite matrices of size $n \times n$. How to prove the positive square root function $\sqrt{\cdot}:\operatorname{Psym}_n \to \operatorname{Psym}_n$ is uniformly continuous? I am quite sure this is true, since on any compact ball this clearly holds, and far enough from the origin, I think the rate of change […]

column space of positive semidefinite matrix

Let be $A$ a symmetric matrix in block form $$A= \begin{bmatrix} B & C \\ C^T & E\end{bmatrix}$$ and let $\operatorname{cs} A$ be the column space for $A$. Because $A$ is positive semidefinite $$\operatorname{cs} \left[\matrix{C^T & E}\right] = \operatorname{cs} E.$$ Why? Thanks for all explanations.

Conditions for positive definiteness: matrix inequality

Let $0<\alpha<1$ and $A,B\in\mathbb{R}^{n\times n}$. I am trying to find conditions on $A$ and $B$ such that \begin{equation} I_n-\frac{1}{\alpha}B^{\rm T}B-\frac{1}{4\alpha(1-\alpha)}( A^{\rm T}B+A)^{\rm T}( A^{\rm T}B+A)>0. \end{equation} However, I do not know how to proceed. Any idea or suggestion is appreciated.

Definiteness of a general partitioned matrix $\mathbf M=\left$

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 semidefinite and indefinite? It is readily seen that $\det(\mathbf M)=\alpha\det(\mathbf A)$, where $\alpha=\mathbf d-\bf b^\top A^{-1}b$ Now, $\alpha>0\Rightarrow \det(\mathbf M)>0$ $\quad(\det(\mathbf A)>0$ by hypothesis$)$ This is not […]

Show that if $X \succeq Y$, then $\det{(X)}\ge\det{(Y)}$

Assume that two symmetric positive definite matrices $X$ and $Y$ are such that $X-Y$ is a positive semidefinite matrix. Show that $$\det{(X)}\ge\det{(Y)}$$ I felt this result is clear, but I can’t use mathematics methods explain this why?

A practical way to check if a matrix is positive-definite

Let $A$ be a symmetric $n\times n$ matrix. I found a method on the web to check if $A$ is positive definite: $A$ is positive-definite if all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column. I couldn’t […]

How to generate random symmetric positive definite matrices using MATLAB?

Could anybody tell me how to generate random symmetric positive definite matrices using MATLAB? Thank you very much for the help and suggestions.

What is the agreed upon definition of a “positive definite matrix”?

In here: A positive definite matrix is a symmetric matrix A for which all eigenvalues are positive. – Gilbert Strang I have heard of positive definite quadratic forms, but never heard of definiteness for a matrix. Because definiteness is higher dimensional analogy for whether if something is convex (opening up) or concave (opening down). […]