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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For any SVD $A = U\Sigma V^T$ of a positive definite, symmetric matrix $A \in \mathbb{R}^{n \times n}$, we have $U = V$.
First of all, I've read through all of the answers here and here, but neither of those threads was able to give completely satisfying answers.
Now, I understand that, if $A$ is symmetric and positive ...
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Show that function is positive definite
In the process of showing that the function\begin{equation}
\langle \mathbf{z}, \mathbf{w}\rangle=\overline{z_1}w_1+(1+i)\overline{z_1}w_2+(1-i)\overline{z_2}w_1+3\overline{z_2}w_2
\end{equation} is ...
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Standard form of positive semidefinite polynomial
The standard form of a positive semidefinite quadratic polynomial of $n$ variables is a sum of n squares. Or in the language of linear algebra, a positive semidefinite symmetric matrix can be ...
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I want to check the positive definiteness of the matrix $\Lambda$
Background
We consider independent and identically distributed (i.i.d.) random variables $$X_1,\ldots, X_n \overset{\text{i.i.d.}}{\sim} N_p(0, \Sigma).$$
Setup
Let $\Sigma$ be a $p$th order positive ...
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singular positive semi-definite matrix in electromagnetism
anyone knows where he drew this conclusion from?
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Verify that a quadratic form is NOT positive definite
Verify that the quadratic form $$q(x_1,x_2,x_3)=x_1^2+4x_1x_2+3x_2^2+2x_2x_3+6x_3^2$$ is NOT positive definite and find a vector in $v\in\mathbb{R}^3$ such that $q(v)<0$ .
I have made several ...
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Coefficients of the characteristic polynomial and positive definite matrices
I revisited my old notes and saw that my former tutor once told us in Linear Algebra that if we want to check if a matrix $\bf A$ is positive definite, then we can check the coefficients of the ...
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Handling the positive square root of the commutator of two positive definite matrices. [closed]
Let $A,B>0$ be non-commuting positive definite operators. Define the commutator to be
$$[A,B] = AB - BA$$
For every operator $A$, $A^*A$ is always positive, and its unique positive square root is ...
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linear algebra - prove that a matrix is positive definite [duplicate]
I'm given a matrix $K$ that is symmetric and positive definite. I'm asked to prove that $K_2 - (l\times u)$ is also a positive definite matrix. $K_2$ is a submatrix of $K$ so that we get rid of the ...
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Proof of Conjecture on `Block-Orthogonalisation Always Reduces Trace'
I have a symmetric positive definite matrix, ${\bf D}\in\mathbb{R}^{NK\times NK}$, which is made up of $K$, $N\times N$, blocks:
\begin{equation}
{\bf D} = \begin{bmatrix}{\bf D_{11}} & {\bf D_{12}...
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Prove symmetric matrix $A$ is congruent to $A ^2$ iff $A$ is PSD [duplicate]
How can I solve the following question:
Let ( A ) be a symmetric matrix. Prove that ( A ) is congruent to
( A^2 ) if and only if ( A ) is positive semidefinite (PSD)
I 've no idea where to start ...
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How to verify positive definitiveness of the given Kinetic term?
I was going through this paper on QCD chaos, where in Appendix B (page 10), for equation B12:
$$\frac{\mathcal{S}}{\mathcal{T}}= \int dt\sum _{n=0,1} \left(\dot{c}_n^2-c_n^2 \omega _n^2\right)+11.3c_0^...
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Smallest eigenvalue of $A^T D^{-1} A + D$ for positive diagonal $D$.
I am wondering whether or not there exists a way to analyze the eigenvalues of
$$A^T D^{-1} A + D$$
for a square matrix $A \in \mathbb{R}^{n \times n}$ and positive diagonal matrix $D\in \mathbb{R}^{n ...
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$\int_{x_i>0} f(x_1+x_2, x_3+x_4)^* f(x_1+x_3, x_2+x_4) \;(x_1x_2x_3x_4)^\alpha \geq0$ for any function $f$ and $\alpha >-1$?
Let $f(u,v)$ be a "nice" complex-valued function defined on the region $u,v\geq 0$, e.g., $f$ is continuous and decays rapidly at infinity.
Problem: For any such function $f$ and for any $\...
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Proving positive semidefiniteness of a matrix multiplication
Assume $Q \in \mathbb R^{n\times n}$ is positive semidefinite and $I$ is the identity matrix. How can I prove that the matrix
$$M = (I + \alpha Q)^{-1} Q $$
is positive semidefinite for all $\alpha \...
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