# 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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### 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: {\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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### Can we prove that ABA' is positive definite if B is positive definite and A is of full row rank

Suppose $A$ is a $r \times n$ matrix and $Rank(A) = r$, $B$ is a $n \times n$ symmetric matrix and $rank(B) = n$. Can we prove that $ABA'$ is positive definite? If it is not positive definiteness, ...
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### Proving inequality regarding expected error involving covariance and its estimate.

Let $$A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota).$$ Where $\Sigma$ is the true $nxn$ covariance matrix of a random vector x that is normally ...
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### Positive definiteness of the derivative of a real-valued positive definite matrix

Is the derivative of a real-valued positive definite matrix is also a real-valued positive definite matrix? If that is not always the case, when it is guaranteed?
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### Making a symmetric matrix positive (semi-)definite by adding a diagonal matrix

Say I have a matrix $A \in R^{p \times p}$ which is symmetric and with non-negative diagonal entries (i.e. $a_{ii} \geq 0 \forall i\in \{1, \ldots, p\}$). However, $A$ is not positive (semi-)definite. ...
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### How to show the quadratic form of the following matrix converges to zero?

Suppose we have a $l\times l$ real matrix $M=(X'X)^{-1}X'\Sigma X (X'X)^{-1}$, where $X$ is a $n\times l$ real matrix and $\Sigma$ is a $l\times l$ real symmetric and positive definite matrix. $X'X$ ...
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### Solving a "quadratic equation" in $\mathbb{R}^n$ with linear constraints on $\mathbb{R}^n$

Problem: Given that $L$ is a subspace of $\mathbb{R}^n$ and denote $L^\perp$ as the orthogonal complement of $L$, i.e. we can write $\mathbb{R}^n = L \bigotimes L^\perp$. Let $w \in \mathbb{R}^n$ be a ...
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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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### Would $v^TAv>0$ forall $v$, $v$ if A's eigenvalues are all > 0? [duplicate]

I'm learning that if $n \times n$ positive definite matrix A A's then $\forall v$ $v^TAv>0 \$, but this requires A is symmetric(then A have orthonormal eigenvectors) so I'm asking: 1.if A is not ...
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### Sum of positive semi-definite matrix and positive definite matrix?

Is the sum of a positive semidefinite matrix and positive definite matrix a positive definite matrix? I have a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$ and the identity $I_n$, is ...
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### how to show that a nonsingular positive semidefinite matrix is positive definite?

Got a question regarding the following statement: a nonsingular positive semidefinite matrix is positive definite. Is this true? If yes, how to show it. If the above statement is not true, how about ...
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### Factorizing $AMA^T+N=WW^T$ efficiently.

This question is related to this question with a slightly more general setting. A good speedup on this could improve performances of a decision process in multi-objective optimization that I designed. ...
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### If $A$ is a symmetric positive definite matrix, show that $f(x) = x^TAx$ is convex.

Let: \begin{gather*} f: \mathbb{R}^n \to \mathbb{R}, \quad A \in \mathbb{R}^{n \times n}, b \in \mathbb{R}^n, x \in \mathbb{R}^n, c \in \mathbb{R} \\ f(x) = x^T A x + b^T x + c \\ \end{gather*} If ...
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### Sum of norms induced by PSD matrices

Suppose you have two positive definite matrices $M$ and $N \in \mathcal{S}_{++}^n$. They induce scalar products and norms on $\mathbb{R}^n$: \begin{align} \lVert x \rVert_M &= \sqrt{x^\intercal ...
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### $0 ≤ A ≤ I.$ $\iff$ ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$

Given two operators $A$and$B$, where $A ≤ B$ means the operator $B − A$ is positive semidefnite. (i) $0 ≤ A ≤ I.$ (ii) ⟨ψ|A|ψ⟩ ∈ [0, 1] for every unit vector $|ψ⟩ ∈ H$ are equivalent I am having ...
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### For an $n$ by $n$ symmetric matrix $M$, why does $Tr(M)^2/Tr(M^2)>n-1$ imply that M is positive or negative definite?

Let $M$ be an symmetric square matrix of size $n$. I am trying to prove that $Tr(M)^2/Tr(M^2)>n-1$ is a sufficient condition for proving that $M$ is positive or negative definite. If in addition \$...