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# Questions tagged [positive-semidefinite]

Relating to a symmetric $n\times n$ real matrix $(M)$ such that the scalar $x^TMx\ge 0\ \forall x\in \Bbb{R}^n\backslash \{0\}$

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### Proof that $\ln \det(\mathbf I + \mathbf X\mathbf Y^{-1}) = \ln \det (\mathbf I + \mathbf Y^{-1/2}\mathbf X \mathbf Y^{-1/2})$

How do we prove that $$\ln \det(\mathbf I + \mathbf X\mathbf Y^{-1}) = \ln \det(\mathbf I + \mathbf Y^{-1/2}\mathbf X \mathbf Y^{-1/2}).$$ Here $\mathbf I$ is the identity ...
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### Is it true that $tr(APA^T)$ > $tr(AQA^T)$, if $tr(P)$ > $tr(Q)$

Assuming $P$ and $Q$ and positive definite matrix. Is it true that $tr(APA^T)$ > $tr(AQA^T)$, when $tr(P)$ > $tr(Q)$. [EDIT after first answer: not just that, actually $P_{ii} > Q_{ii}$ for ...
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### singular positive semi-definite matrix in electromagnetism

anyone knows where he drew this conclusion from?
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### Limit of $\mathrm{tr}(P(\alpha QP + I)^{-1})$ as $\alpha \to \infty$?

Say that $P, Q$ are two real, symmetric positive semidefinite, possibly singular matrices. What is the limit $$\lim_{\alpha \to \infty} \mathrm{tr}(P(\alpha QP + I)^{-1})?$$ Simultaneously ...
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### Why the first matrix is not always psd but the second matrix is always psd?

I learned that one important motivation for the Newey-West covariance estimator is that naive estimator of the covariance matrix is not necessarily positive semidefinite(psd from now on) (see the ...
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### Trace of product of positive semi definite matrix and matrix with all negative eigenvalues

Suppose $A \in \mathbb{R}^{n \times n}$ is a Hurwitz matrix, that is, all the eigenvalues of $A$ have strictly negative real parts, and that $S \in \mathbb{R}^{n \times n}$ is a symmetric positive ...
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### Example of semi algebraic subset over a real closed field

I am trying to find interesting example of semi algebraic subset over a real closed field other than those we have by considering $\mathbb{R}^n$. I tried to construct an example involving matrices. ...
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2 votes
1 answer
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### Trace of symmetric part of product of 3 positive semidefinite matrices

Suppose we have 3 symmetric positive semi-definite matrices $A,B,C$. How can one prove or give counter-example for the following statement? $$\mathrm{trace}(AB(A-C)^2 + (A-C)^2BA) \ge 0$$ where we ...
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### Operator norm of the sum of two positive semidefinite matrices

Consider two positive semidefinite and symmetric matrices, namely $\bf{A}$ and $\bf{B}$. Denote $\Vert\cdot\Vert$ as the operator norm. If we have another positive semidefinite matrix $\bf{B}^*$ that ...
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### What is the reason for endowing the positive semidefinite cone with the Affine Invariant Riemannian Metric instead of a Euclidean metric?

I came across this post Distances defined in manifold of symmetric positive definite matrices because I had the same questions. I did not understand the answers provided, so I wanted to try to answer ...
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### subset of positive matrices such that the Loewner order is total

Are there known characterizations of subsets of positive matrices over which the Loewner order is total ?
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### Limit of sum of factions of finite constants and matrix

Suppose $\mathbf{a}_{i}$ is a $2\times 1$ constant vector with $\frac{1}{N}% \sum_{i=1}^{N}\mathbf{a}_{i}\mathbf{a}_{i}^{T}$ converges to a finite $% 2\times 2$ matrix as $N\rightarrow \infty$, and \$...
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