Questions tagged [positive-definite]

For questions about positive definite real or complex matrices. For questions about positive semi-definite matrices, use the (positive-semidefinite) tag.

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$A$ is positive definite iff $\det(A_k) > 0$

Let $A$ be a symmetric $n \times n$ matrix, and $V = \mathbb{R}^n$. Define $\langle v,w \rangle = v^t A w$ with $v,w \in V$. Show that $A$ is positive definite iff $\det(A_k) > 0$ for $1 \leq k \...
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Invertibility of positive semidefinite matrice [closed]

Excuse my boldness in asking such a basic question, this holds the key to my understanding of a certain concept: Is a positive semi-definite matrix invertible $\mathit{iff}$ it is non-singular? ...
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14 views

A positive definite hermitian form induced by killing form

Let L be a simple Lie algebra with chevalley generators $\{x_i, y_i, h_i | i=1,...,l\}$. Then there is an unique antilinear anti involution $\omega_0 : L\to L$ s.t. $$\omega_0 x_i=-y_i \\ \omega_0 y_i=...
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1answer
41 views

Values that make matrix positive definite

Assume the symmetric matrix: $$ M= \begin{bmatrix} \frac{\sigma\omega\pi^2}{4L^2} + g & 0 & -\frac{\sigma + c^2Qg}{2\sigma} \\ 0 & a\mu & -\frac{Q}{2}\left(\frac{c^2}\sigma +...
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How to prove that this block-matrix is positive-definite?

I have a $3n\times3n$ symmetric block matrix that I need to prove is positive definite: $$ M = \left(\begin{array}{ccc} M_{1,1}&\dots&M_{1,n}\\ \vdots&\ddots&\vdots\\ M_{n,1}&\...
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29 views

Formula for eigendecomposition of symmetric matrix

In some books/notes, the eigendecomposition of positive definite matrix $\bf A$ is written as \begin{align*} {\bf A} = {\bf P}^{\bf T}{\bf \Lambda}{\bf P} \Longrightarrow {\bf \Lambda} = {\bf P}{\bf ...
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1answer
29 views

Prove that a particular matrix is positive definite

Let $A$ and $V$ be positive semidefinite matrices, and let $U$ be an orthogonal matrix, partitioned columnwise: $$U = \begin{bmatrix} U_1 & U_2 \end{bmatrix}$$ Consider the matrix $$M = A + U_1^...
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1answer
46 views

If A positive definite then $\lvert\det(A)| \leq \prod_{i=1}^n \Vert a_{i}\Vert$

I've asked a similar problem earlier here and I was told the same is true for $\lvert\det(A)| \leq \prod_{i=1}^n \Vert a_{i}\Vert$ with $\Vert a_{i,i}\Vert$ being the norm induced by the Euclidean ...
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1answer
80 views

If A positive definite then $\det(A) \leq \prod_{i=1}^n a_{i,i}$

If $A=(a_{ij})_{ij=1,...n}\in \mathbb{C}^{n \times n}$ positive definite $ \Rightarrow \det(A) \leq \prod_{i=1}^n a_{i,i}$ I've argumented with the Cholesky-decomposition (which exists because A is ...
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Question about finding the open set where the matrix is positive definite using the characteristic polynomial.

Let $H := \begin{pmatrix} \gamma_1 & \delta \\ \delta & \gamma_2\end{pmatrix}$. I want to find the open set $C \in \mathbb R^3$ such that for every $(\gamma_1,\gamma_2,\delta)\in C$ the inner ...
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1answer
11 views

Spectral radius of $B$ if $W-B^TWB$ is positive definite

Problem: Suppose that $W = S^TS$ for some square matrix $S$, and that $W-B^TWB$ is positive definite. Show that the Spectral Radius of $B$ is less than $1$. Attempt: $W = S^TS$ is symmetric, so ...
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1answer
84 views

Product of matrices has real eigenvalues?

Let $A$ be a (symmetric) positive definite matrix and $\hat{n}$ be an arbitrary unit vector. Consider $b,c,d,k$ arbitrary positive integers. I would like to know if the following matrix has real ...
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6 views

Iterative tests for positive-definiteness in large scale setting

Given a symmetric linear function $L\colon\mathbb R^N \to \mathbb R^N$, $N\ggg 0$, do there exist (possibly probabilistic) iterative tests for positive definiteness? Ideally I am looking for an ...
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1answer
17 views

Find a symmetric positive-definite matrix

I need to find a symmetric matrix A \begin{pmatrix} a_{11} & a_{12} & a_{13}\\ a_{12} & a_{22} & a_{23}\\ a_{13} & a_{23} & a_{33} \end{pmatrix} such that its' main minors are ...
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1answer
30 views

Prove that Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k

I need some help, I need to prove the following exercise: Let A be a definite positive matrix, such that for every positive integer k, there exists a symmetric matrix B such that A=B^k I haven't ...
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54 views

Condition for non-negative Fourier transform: $\mathcal{F} \exp(-|x|^a)(|x|^a\log(x)+\delta) \ge 0$

I am looking for $\delta>0$, such that Fourier transform of $\exp(-|x|^a)(|x|^a\log(x)+\delta)$, $a \in [1,2]$ is non-negative. I have tried to evaluate the integral for a=2. I failed to arrive at ...
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2answers
80 views

Prove or disprove $\tfrac{b^{\top}Ab}{b^{\top}A^{-1}b}\leq\tfrac{\lVert Ab \rVert^{2}}{\lVert b \rVert^{2}}$ [closed]

Let $A$ be positive definite and symmetric and $b\neq 0$. Is $\tfrac{b^{\top}Ab}{b^{\top}A^{-1}b}\leq\tfrac{\lVert Ab \rVert^{2}}{\lVert b \rVert^{2}}$ true? Any idea how to prove it, maybe with ...
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1answer
95 views

Relation between $\infty$-norm and 2-norm condition number of PD matrix

Consider a positive-definite matrix $A$ in $\mathrm{R}^{n\times n}$ and let $\kappa_{\infty} = \|A\|_{\infty}\|A^{-1}\|_{\infty}$ and $\kappa_2 = \|A\|_2 \|A^{-1}\|_2$, with $\|A\| _p = \sup_{x \ne 0} ...
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Solving generalized eigenvalue problem with almost positive definitive matrices?

Assume that we are going to solve: $$AV = BVD$$ It's the generalized eigenvalue problem. In this case, $A$ and $B$ are symmetrical. But they are not positive definitive because most of its ...
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22 views

If $X^T$ is full rank, then $X^TX$ is positive definite?

I read that If $X^T$ is full column rank ($X$ is not necessarily square), then $X^TX$ is positive definite, and I'm trying to see why this is the case. I know that if $X^T$ is full column rank, then $...
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19 views

Show that there exist matrices $A$ and $B$ such that this matrix series is not positive semi-definite

This is a follow-up on a question I asked before. Suppose that $A$ is a $k \times k$ matrix with $\|A\|_2 \leq 1$ and $B$ is a $k \times k$ positive semi-definite matrix. Consider the following sum: ...
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1answer
28 views

Positive Definiteness of a symmetrized matrix involving the Moore Penrose Pseudoinverse

Let $M\in\mathbb{R}^{m\times n }$ and $V\in\mathbb{R}^{n\times l }$ both have full column rank and let $X^{+}$ denote the Moore Penrose Pseudoinverse of the matrix $X$. Question: Is the symmetrized ...
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31 views

Given a matrix $A \in \mathbb{R}^{n \times n}$, $A + A^T$ is positive (semi)definite, what is $A$?

Given a matrix $A \in \mathbb{R}^{n \times n}$, $A + A^T$ is positive (semi)definite, what is $A$? or more generally, what are the properties on $A$ such that it holds true? We see that if $A$ is ...
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1answer
43 views

Proof of positive semi-definiteness of hat matrix product

Consider the matrices $B = \begin{bmatrix}1 &0\\-1&1\\0&1\end{bmatrix}$, $Y = \begin{bmatrix}x_1&0 & 0\\0&x_2&0\\0&0&x_3 \end{bmatrix}$, $\tilde{B} = YB$ and $T =...
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1answer
21 views

Positive- definiteness of factors of Cholesky factorization

If A is a symmetric positive-definite n×n matrix, then is the lower triangular n×n matrix L positive-definite when A=LL* using Cholesky factorization?
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18 views

Mean orthogonal projection on a vector of multidimensional normal distribution

Let $X\in\mathbb{R}^n$ be a normally distributed random vector with mean zero and positive definite covariance matrix $\Sigma$, i.e. $X\sim\mathcal{N}(0,\Sigma)$. Treating $X$ as a column vector, the ...
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15 views

Apportionate positive and negative value

I have set of values 5,8,10,12 and i have another adjustment value 20. i wanted adjustment has to be apportioned to other set of values. I did like 5+8+10+12=35. 5/35*20 for 5 and likewise for rest ...
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1answer
33 views

Show that this matrix series is positive semi-definite

$A = B^\top B$ is a $k \times k$ symmetric positive semi-definite matrix and $\|A\|_2 \leq 1$. Consider the following sum for $c$: $$S_n = \sum_{i=0}^{n-1} (n-i) A^{i} - \frac{1}{c} \sum_{i=0}^n A^{i}...
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1answer
30 views

Is a symmetric matrix positive definite iff $D$ in its LDU decomposition is positive definite?

Given $$A=LDU$$ where $A$ is a real symmetric matrix $L$ is a lower unitriangular matrix $D$ is a diagonal matrix $U$ is an upper unitriangular matrix can we say that $$A>0 \iff D>0$$ ? Edit:...
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61 views

Determining whether a matrix is positive definite from its LU decomposition

Given that $A=LU$ where $L$ and $U$ are (known) lower and upper triangular matrices, is there any simple way to determine whether $A$ is positive definite? Background I have been using this ...
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2answers
81 views

Is there any obvious way to enforce a minimum amount of “positive definiteness” on a matrix?

Let $f(A,F)=\max(A,F)$ where $A\in\mathbb{R}$ is a variable and $F\in\mathbb{R}$ is a constant representing a "floor" below which the result should not be permitted to go. Is there any obvious ...
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28 views

Properties of a symmetric, positive definite matrix

Consider symmetric, positive definite matrix $B \in \mathbb{R^{n \times n}}$. Show that the following holds $$|b_{ij}| \leq \sqrt{b_{ii} \cdot b_{jj}} \leq \frac{1}{2} (b_{ii} \cdot b_{jj}) $$ How ...
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1answer
21 views

Proving definiteness of matrix implicates the sign of diagonal

Let $A\in \mathbb R ^{n\times n}$ be positive definite. Prove that this implicates that all elements on the main diagonal are positive. So $A$ is positive definite means, that $v^T Av>0$, where $v^...
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1answer
39 views

Show that $A$ is positive definite via the Cholesky decomposition

I have calculated the Cholesky decomposition of the matrix \begin{equation*}A=\begin{pmatrix}1 & -1 & -1 & 0 \\ -1 & 5 & 5 & -4 \\ -1 & 5 & 6 & -3 \\ 0 & -4 &...
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1answer
32 views

Rank-1 random matrix is positive definite?

Let $X\in\mathbb{R}^n$ be a random vector with continuous coordinates such that $\|X\|_2=1$ a.s. Define the random matrix $A=XX^T$. It is obvious that $A$ has rank 1 and its unique non-zero eigenvalue ...
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1answer
24 views

Positive definitness of the expectation of a random matrix

Let $X$ be a non-negative random variable. We know that if $E[X]= 0$ then $X=0$ a.s. A consequence of this is that $P(X\neq 0)=1$ implies $E[X]>0$. Now, let $A$ be a $n\times n$ symmetric and non-...
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Does $\sum_i A_i=I$ with $A_i$ positive imply $\{A_i\}_i$ are mutually diagonalisable?

As discussed in this other question, if $A$ and $B$ are matrices such that $A+B=I$, then trivially they commute, and thus if they are both diagonalisable they are also mutually diagonalisable. The ...
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1answer
58 views

Matrix Eigenvalue/Positive definiteness optimization problem

I have encountered a matrix optimization problem from a system of ODE's, however I am not very familiar with optimization. Let $A(\lambda)$ be a $4 \times 4$ matrix dependent on a value $\lambda >...
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1answer
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Conditions for Eigenvalues of a Self-Adjoint Endomorphism to Lie in $[a,b]$

It is well known that for an endomorphism $\alpha: V \to V$ with $V$ a finite-dimensional inner-product space over $\mathbb{C}$, $\alpha$ has purely real eigenvalues. But if we want to bound these ...
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1answer
24 views

Which precomputations can be performed to invert product of positive symmetric matrices faster?

The problem Given two real symmetric positive definite matrices $\Sigma_1$ and $\Sigma_2$, and real positive diagonal matrices $A_1$ and $A_2$, I want to compute the inverse of the product $$ C = ...
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1answer
39 views

How to quickly determine the definiteness of a large sparse matrix without using Sylvester's criterion?

I am currently trying to classify the stationary points of a function as either a maximum, minimum or saddle points based on the definiteness of the Hessian at those points. I have worked out that ...
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0answers
27 views

Proof for positive definiteness of a given matrix

Let A be N$\times$ N symmetric matrix having all real entries. The entries of A be $a_{ij}$ where $i,j \in {1,2,..N}$ and $a_{ij}$ is given by $$a_{ij}=\begin{cases} f(n)=\dfrac{n^\zeta-2(n-1)^\zeta+...
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19 views

Is the function $V=x^{2}$ a positive semi-definite function?

I recently started studying The Mathod of Lyapunov, and I thought I understood how to distinguish a positive definite from a positive semi-definite functions, or negative from semi-definite functions. ...
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24 views

To find class K infinity bounds on given radially unbounded function

Let $f = x_1^2 + x_2^4$. How to find class $\mathcal{K_{\infty}}$ functions $\alpha_1(||x||)$ and $\alpha_2(||x||)$, such that, $\alpha_1(||x||) \leq f \leq \alpha_2(||x||), \forall x \in {R}^2.$ $\...
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1answer
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Finding conditions for an inequality, positive semi definite

I want to prove that the sequence: $$ ( 2 + \psi^2 ; \psi ) $$ is positive semi definite for $|\psi | < 1$, and ideally find conditions on $\psi$ about when it isn't. Recall that for a sequence, ...
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1answer
49 views

proving that lu decomposition is not unique on singular matrix.

How to prove that the following isn't true (using 3 by 3 matrix): Given A is a square and a singular matrix (which means non invertible), if LU decomposition is possible without the use of ...
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32 views

Question about the diagonalizability

In some textbooks it is mentioned that, if a matrix is Positive Semidefinite, then all its diagonal entries are nonnegative and all the principal submatrices of PSD obtained by removing any number of ...
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51 views

Does positive definiteness imply $x^{T}Mx \geq \alpha x^{T}x$ for some $\alpha > 0$?

I require an $n \times n$ matrix to satisfy $x^{T}Mx \geq \alpha x^{T}x$ for some $\alpha > 0$ for all $x \in \mathbb{R}^{n}$ where $x \geq 0$ (1). Is it sufficient to show that M is positive ...
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1answer
58 views

The number of positive eigenvalues of a non-singular symmetric matrix is equal to the number of positive pivots(Strang)

I am reading "Introduction to Linear Algebra 4th Edition" by Gilbert Strang. In this book, there is a theorem which says the number of positive eigenvalues of a non-singular symmetric matrix is the ...
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2answers
21 views

Positive definite implies positive semi-definite

I have read in the post, but there is something missing I'd like to know. I do believe it's not a duplicate post. I have a specified question, probably a trivial one. First, let $(s_n)_{n\geq 0}$ be ...

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