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 […]

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 […]

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 […]

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.

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.

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 […]

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?

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 […]

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

In here: http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/positive-definite-matrices-and-applications/symmetric-matrices-and-positive-definiteness/MIT18_06SCF11_Ses3.1sum.pdf 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). […]

Intereting Posts

The Relative Values of Two Modified Bessel Function of the First Kind
Circle enclosing all but one of $n$ points
Prove that in a sequence of numbers $49 , 4489 , 444889 , 44448889\ldots$
Order of cyclic groups and the Euler phi function
Subset $A\subset\mathbb R$ such that for any interval $I$ of length $a$ the set $A\cap I$ has Lebesgue measure $a/2$
Irreducible solvable equation of prime degree
Is it possible to imitate a sphere with 1000 congruent polygons?
How can apply the $L^p$ norm in a circle to $L^2$ norm in a square?
Explanation of proof of Gödel's Second Incompleteness Theorem
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
A Combinatorial Proof of Dixon's Identity
Intuition about infinite sums
Picture of a 4D knot
How many lists of 100 numbers (1 to 10 only) add to 700?
Map of Mathematical Logic