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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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44 views

Is $A^2-B^2$ positive definite too when $A-B,B$ is positive definite?

Denote $A,B\in M_n(\mathbb{R})$ If $A-B,B$ is positive definite, it's easy to see $A^2-B^2$ is symmetric. Now the question is: Prove or disprove: $A^2-B^2$ is positive definite. I have checked ...
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0answers
32 views

Question of the Cholesky decomposition of symmetric positive definite matrix

This is a exercise on my numerical analysis textbook: Suppose $\mathbf A$ is a positive-definite symmetric matrix, and the Cholesky decomposition is of the form $\mathbf {A} =\mathbf {LL}^{T}$, ...
0
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1answer
26 views

Is the trace of the matrix obtained by subtracting two positive definite matrices smaller than the trace of the matrix being subtracted?

Let $A$, $B$ be symmetric positive definite matrices. If $C=A-B$, is it true that $$ \operatorname{trace}\left(C^{-1}\right) > \operatorname{trace}\left(A^{-1}\right)\:?$$
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19 views

Monotonicity of matrix inverse of positive definite matrices [duplicate]

If $A$ and $B$ are positive definite, $A-B$ is positive definite, can we say $B^{-1}-A^{-1}$ is positive definite? I think this should be true but I don't know how to prove it. It would be great if ...
1
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3answers
69 views

Showing matrix $\left[\begin{smallmatrix} 4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3 \end{smallmatrix}\right]$ is positive definite

$$\begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix}\begin{bmatrix} 4 & 1 & 1\\1 & 2 & -1 \\ 1 & -1 & 3 \end{bmatrix}\begin{bmatrix} x_1\\x_2\\x_3 \end{bmatrix} =...
1
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1answer
10 views

Showing that the positive operator absolute value satisfies a certain inequality

Let $H$ be a Hlbert space. If $T$ is a bounded linear operator on $H$, then $|T|=\sqrt{T^*T}$ is called the absolute value of $T$. And if $A$ and $B$ are self-adjoint bounded linear operators, then ...
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0answers
27 views

Can a symmetric positive definite tridiagonal matrix have a zero coefficient on its diagonal?

I'm investigating the Parallel Cyclic Reduction algorithm (Fast Tridiagonal Solvers on the GPU) in the case of positive definite tridiagonal matrices and I'm wondering whether a 0 can appear on the ...
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0answers
46 views

Can this matrix be negative definite?

Let $d = 12$ and $m = 6$, and denote by $0_n$ and $I_n$ the zero matrix and the identity matrix of size $n \times n$. Let $D_+ \in \mathbb{R}^{m \times m}$ be a diagonal matrix with positive ...
0
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1answer
21 views

Positive definite in mean square sense

$x$ is a random vector variable and matrix $P$ is symmetric and deterministic. If $\mathbb{E}\{x^T P x\}>0$ for any non-zero $x$, can we say $P$ is positive definite? If it is not, can you give me ...
2
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0answers
36 views

Non-singularity of a certain block matrix

We are given a matrix of the form $$\left[\begin{array}{cccccccccccccc} 0&0&0&0&0&0&0&0&0&q_{1}&0&1&0&0\\ 0&2p_{2}q_{2}&0&0&...
0
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2answers
37 views

Necessary and sufficient conditions for rank-$1$ update of positive definite matrix to be positive definite

Given $A \succ 0$, what are necessary and sufficient conditions on $\lambda \in \mathbb{R}$ for $A + \lambda u u^\top$ to be positive definite? What happens if we relax $A \succeq 0$ and require $A + ...
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4answers
81 views

Find matrix $A$ given the matrix $X$ and that $X = AA^T$

I have a matrix $X = \begin{bmatrix}3 & 1\\1 & 1\end{bmatrix}$ and $X=AA^T$. How can I find $A$?
0
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1answer
25 views

Product of positive definite matrix and matrix symmetrization

Suppose $X$ and $A+A^*$ are positive definite matrices. I would like to ask if it is true that $AX+ (AX)^* = AX + XA^*$ is also positive definite?
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22 views

Trace of product of squared matrix and positive definite matrix is nonnegative (short exercise)

Let $\Sigma$ be a real $d \times d$-matrix and $C$ be a real, symmetric, positive definite $d \times d$-matrix. Does it then hold, that $$ \text{tr}(\Sigma^2C) \geq 0 \quad ?$$
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0answers
42 views

positive definite function

Let $\phi = [x1^2, x1x2, x2^2]^\top$. $W_1 ^\top \phi(x)$ and $W^{* \top} \phi(x)$ are positive definite functions and $A$ is a rank-1 positive semi definite matrix ($0<$one eigenvalue$<1$, the ...
1
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1answer
13 views

when x'y >0 and A is positive definite matrix, x' A y >0?

when $x' y >0$ and A is a positive definite matrix, $x' A y > 0$? Here, x and y is column vector so that $x' y$ is scalar.
2
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1answer
37 views

Stationary point of $(x_2 - x_1^2)^2 + x_1^5 $not local max/min

I need to prove that $f(x)=(x_2 - x_1^2)^2 + x_1^5 $ has one stationary point which is neither a local max nor min The stationary point of $f(x)$ is found by $\nabla f(x)=0$ which gives $x_1 = x_2 = ...
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0answers
9 views

Sylvestor criteria in terms of leading principal sub-matrices

It is known that the leading principal (NORTH WEST)submatrices of a positive definite matrix are positive definite. But can this be considered as a necessary condition for positive definiteness (...
2
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2answers
47 views

Prove that $f$ is strictly convex iff $A$ is positive definite

I would like to have a help with this question. Let $$f(X) = \frac{1}{2}X^TAX + b^TX$$ with $A=A^T$. Prove that $f(X)$ is strictly convex if and only if $A$ is positive definite. Regards!
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1answer
59 views

Determinant of a positive semi-definite matrix

If $M$ is a Hermitian matrix, then $M$ is positive definite if and only if its leading principal minors have positive determinant, i.e the following matrices have positive determinant: The upper-left ...
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1answer
44 views

Solving a system by using Cholesky Decomposition $(LDL^T)$

$$\begin{bmatrix} 4 & 1 & 1 \\ 1 & 2 & -1 \\ 1 & -1 & 3 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 3 \\ ...
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0answers
37 views

Why is positive definiteness required for a global minimum to exist?

When we optimize $$\min x'Ax$$ why does matrix $A$ need to be positive definite in order to have a global minimum?
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1answer
39 views

After multiplying a positive definite matrix several times to 'a vector A', still less than 90 degree between the 'vector A' and the 'mapped vector'?

My question Would the $\theta$ be still less than 90 degrees in vT * Mk v = ||v|| * ||Mk v|| * cos $\theta$, if the matrix M is positive definite? Background Information Let's suppose that v (...
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1answer
12 views

Showing $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ [closed]

Similar as the title, how to prove that $\Omega = PP'$ implies $P^{-1}\Omega(P')^{-1} = I$, where $\Omega$ is a symmetric and positive definite matrix.
0
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1answer
35 views

Sum of two positive denfinite matrices invertible — where is my mistake?

I wrote the following statement: $A(t) = \sum_{i \le t} r(i) r(i)^\top + \alpha I_N$ where $r(i) \in \mathbb{R}^N $. As a sum of positive definite matrices $r(i)r(i)^\top$ and a ...
4
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1answer
91 views

Let $A, B$ be two positive definite $2 \times 2$ matrices. Prove or disprove: $AB+BA$ is positive definite.

I know that $AB+BA$ is not necessarily positive definite, as this question has been asked before on here. What I don't understand is how one would go about constructing counter-examples. Previous ...
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0answers
28 views

Intermediate matrices in the positive definite ordering.

Let $$ M= \left[ {\begin{array}{cc} a & b \\ b & c \\ \end{array} } \right],$$ be a positive definite matrix such that $M \succeq xI_d$, for some $x > 0$. Here '$\succeq$' means ...
1
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1answer
32 views

Show that DFP update preserve Positive Definiteness?

The update for Davidon-Fletcher-Powell (DFP) is given as the following: $$ B_{k+1} = (I - \rho_ky_ks_k^{\top})B_k(I - \rho_ks_ky_k^{\top})+\rho_ky_ky_k^{\top} $$ where $y_k,s_k \in \mathbb{R}^n$ ...
0
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1answer
75 views

Is there a fast way to compute the lowest eigenvalue of this symmetric PD matrix in this specific scenario?

Consider $$C = A^H D A + M$$ where $A$ is a $m \times m$ unitary matrix. $D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$. $M$ is a $m \times ...
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0answers
28 views

Showing a function $f \colon \mathbb{R}^{d \times d} \times \mathbb{R}^d \to \mathbb{R}$ is in $C^1$.

I have a function $f \colon \mathbb{R}^{d \times d} \times \mathbb{R}^d \to \mathbb{R}$ defined by $$ f(A, c) = (z - c)^T A (z - c) - 1 $$ where $A \in \mathbb{R}^{d \times d}$ is symmetric ...
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0answers
34 views

Positive semi-definiteness?

One of my homework solutions stated the following: However, I do not seem to grasp why the solutions refers to 'both principal minors'. As far as I know there are three principal minors: Order 1: ...
0
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1answer
25 views

Definite positive matrix

According to a book that I have an hermitian matrix is definite positive if $X^TA\overline{X}>0$, but here in the forums (and other sources) the definition is given by $X^*AX>0$. Somehow I'm ...
2
votes
2answers
92 views

Show a specially defined matrix is positive definite

Let $E_1, ..., E_n$ be non empty finite sets. Show that the matrix $A = (A_{ij})_{1 \leq i, j \leq n}$ defined by $A_{ij} = \dfrac{|E_i \cap E_j|}{|E_i \cup E_j|}$, is positive semi-definite. This is ...
4
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3answers
140 views

How to solve $A^{\frac 12} B A^{\frac 12} = C$ for $A$?

Suppose that matrices $A,B,C$ are symmetric and positive definite. Then, $A$ has a unique, positive square root, which we call $A^{\frac 12}$. If $$A^{\frac 12} B A^{\frac 12} = C$$ then can we write ...
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0answers
24 views

The necessary and sufficient conditions on a diagonal matrix $D$ in other that for any $A$, the positive definiteness of $A$ implies that of $DA$?

The question did not clarify which definition of semidefiniteness I should use. If we use the definition that $x^*Ax>0$, then $A$ is symmetric and all the elements of $D$ should be the same ...
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0answers
20 views

Bounds on the support of a correlation matrix' eigenvalue spectrum

I have a covariance matrix $V=(v_{ij})$ and construct a correlation matrix $C$ with entries $c_{ij}=\frac{v_{ij}}{\sqrt{v_{ii}v_{jj}}}$. The matrices $V$ and $C$ are positive definite, so I know that ...
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3answers
37 views

Positive Definiteness of Band Matrix

Let $A$ be the $n \times n$ matrix: $$ A = \begin{bmatrix} a & 1\\ 1 & a & 1\\ &1 & a &1\\ &&\ddots &\ddots & \ddots\\ &&&1 & a& 1\\ &&...
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1answer
45 views

Sufficient conditions for Loewner ordering of two matrices?

Le $\mathbf{A}$ and $\mathbf{B}$ two psd matrices of size $n$. Additionally we assume that the entries are real and non-negative. Does the following hold: $$\forall (i,j) \in [n], \mathbf{A}_{ij} \leq ...
0
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1answer
32 views

Positive definite matrix properties

I am having trouble solving a property that I found. If $A:n \times n$ is defined as a positive definite matrix and $B: n \times m$ where $rank(B) = r$. Then $B^T A B > 0$, only when r = m and $B^...
0
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1answer
58 views

Why is this matrix necessarily positive definite?

A recently asked question here was solved with the claim that any symmetric square matrix $M$ of the following form is positive definite: All of the off-diagonal elements are the same positive ...
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0answers
12 views

Kernel on Grassmannian

The Binet-Cauchy kernel on Grassmannians given in terms of principal angles $\theta_i$ is $K_{bc}^2 := \prod_i \cos^2(\theta_i)$, and is known to be positive definite. However, $L_{bf} := \prod_i \cos(...
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0answers
20 views

Positive definiteness and eigenvalue ellipses

if non-symmetric square matrices A and B holds the following inequality: A $\ge$ B Can I say that matrix A's eigenvalue ellipse encloses that of B's in case of 2D? By eigenvalue ellipse, I mean an ...
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3answers
29 views

Expanding this completing the square problem.

The following is the final stage of a completed proof, which arrived at a completing the square problem that I'm having trouble figuring out the deriving steps. Could anyone help to show the steps of ...
1
vote
1answer
55 views

Weighted inner product with arbitrary matrix?

An inner product can be written in Hermitian form $$ \langle x,y \rangle = y^*Mx $$ that requires $M$ to be a Hermitian positive definite matrix. I have read that using Hermitian positive definite ...
0
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1answer
21 views

Degrees of freedom of the set of positive definitive matrices

If I am not wrong, the set of definite positive matrices with real coefficients is a convex cone without the vertex, which is the null matrix. What is the number of degrees of freedom for this set of ...
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0answers
33 views

Is this matrix defined from an integral of a non-negative function positive definite?

I have a function $f(x)$ defined as \begin{equation} f(x) := \sum_{n = 1}^N c_n f_n(x) = \mathbf{f}(x)^T \mathbf{c}, \end{equation} where \begin{align} \mathbf{f}(x) := \begin{bmatrix} f_1(x) \\ ...
0
votes
1answer
14 views

using one vector to build positive definite matrix

Let $X=(X_1,X_2,...,X_n)_{1\times n}$ be a n-dimensional vector and a matrix $A_{n\times n}=(X^{T})_{n\times 1} * X_{1\times n}$. Under what condition of $X$, $A$ is a (semi-)positive definite matrix? ...
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4answers
55 views

How to show that this matrix is positive semidefinite?

Using the definition, show that the following matrix is positive semidefinite. $$\begin{pmatrix} 2 & -2 & 0\\ -2 & 2 & 0\\ 0 & 0 & 15\end{pmatrix}$$ In other words, if ...
2
votes
1answer
55 views

Inequality for trace of product of matrices

Assume that $A \in \mathbb{R}^{n \times n}$ is a symmetric matrix and $B \in \mathbb{R}^{n \times n}$ is a symmetric positive definite matrix. Is the following statement true $$ \lambda_{\mathrm{min}} ...
1
vote
1answer
31 views

How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?

I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...