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.

This tag is for questions about positive definite matrices with real or complex entries. A square matrix $M \in \mathbf{F}^{n \times n}$ ($\mathbf{F} = \mathbf{R}$ or $\mathbf{C}$) is positive definite if $$ \text{for all } x \in \mathbf{F}^n \setminus \{0\}, x^\dagger M x > 0. \tag{1}$$ Here $x^\dagger$ denotes the transpose if $x$ is real and the conjugate-transpose if $x$ is complex.

If we replace $(1)$ with $$ \text{for all } x \in \mathbf{F}^n, x^\dagger M x \ge 0 $$ then $M$ is said to be positive semi-definite. All positive definite matrices are positive semi-definite. Questions about positive semi-definite matrices not specifically about positive definite matrices should use the tag instead or in conjunction.

If $\mathbf{F} = \mathbf{C}$ then $M$ is positive definite if and only if $M^\dagger = M$ and every eigenvalue of $M$ is a positive real number. If $\mathbf{F} = \mathbf{R}$ then it is not necessary that $M^\dagger = M$, for instance $$ M = \begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix} $$ is a positive definite real matrix but not symmetric. Some authors require that a positive definite matrix be symmetric.

Some authors use a weaker form of $(1)$, namely $$ \text{for all } x \in \mathbf{F}^n \setminus \{0\}, \operatorname{Re}(x^\dagger M x) > 0. $$ With this definition it is no longer necessary that $M^\dagger = M$, even if $\mathbf{F} = \mathbf{C}$.

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If eigenvalues are positive, is the matrix positive definite?

If the matrix is positive definite, then all its eigenvalues are strictly positive. Is the converse also true? That is, if the eigenvalues are strictly positive, then matrix is positive definite? Can you give example of $2 \times 2$ matrix with…
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Does non-symmetric positive definite matrix have positive eigenvalues?

I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues. Does this hold for non-symmetric matrices as well?
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How to generate random symmetric positive definite matrices using MATLAB?

Could anybody tell me how to generate random symmetric positive definite matrices using MATLAB?
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Inverse of a Positive Definite

Let K be nonsingular symmetric matrix, prove that if K is a positive definite so is $K^{-1}$ . My attempt: I have that $K = K^T$ so $x^TKx = x^TK^Tx = (xK)^Tx = (xIK)^Tx$ and then I don't know what to do next.
diimension
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Is there a fast way to prove a tridiagonal matrix is positive definite?

I' m trying to prove that $$A=\begin{pmatrix} 4 & 2 & 0 & 0 & 0 \\ 2 & 5 & 2 & 0 & 0 \\ 0 & 2 & 5 & 2 & 0 \\ 0 & 0 & 2 & 5 & 2 \\ 0 & 0 & 0 & 2 & 5 \\ \end{pmatrix}$$ admits a Cholesky decomposition. $A$ is symmetric, so it admits a…
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Does a positive definite matrix have positive determinant?

Let $A$ be a positive-definite real matrix in the sense that $x^T A x > 0$ for every nonzero real vector $x$. I don't require $A$ to be symmetric. Does it follow that $\mathrm{det}(A) > 0$?
user168931
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When does the inverse of a covariance matrix exist?

We know that a square matrix is a covariance matrix of some random vector if and only if it is symmetric and positive semi-definite (see Covariance matrix). We also know that every symmetric positive definite matrix is invertible (see Positive…
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A practical way to check if a matrix is positive-definite

Let $A$ be a symmetric $n\times n$ matrix. I found a method on the web to check if $A$ is positive definite: $A$ is positive-definite if all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute…
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Do positive semidefinite matrices have to be symmetric?

Do positive semidefinite matrices have to be symmetric? Can you have a non-symmetric matrix that is positive definite? I can't seem to figure out why you wouldn't be able to have such a matrix, but all my notes specify positive definite matrices as…
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The product of two positive definite matrices has real and positive eigenvalues?

Given two real positive definite (and therefore, symmetric) matrices $A$ and $B$, are all the eigenvalues of $AB$ real and positive? Wikipedia says $AB$ is positive definite if $A$ and $B$ are positive definite and commute, but I don't need $AB$ to…
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Characterization of positive definite matrix with principal minors

A symmetric matrix $A$ is positive definite if $x^TAx>0$ for all $x\not=0$. However, such matrices can also be characterized by the positivity of the principal minors. A statement and proof can, for example, be found on…
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Determining whether a symmetric matrix is positive-definite (algorithm)

I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition. The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky decomposition, must be symmetric and positive-definite.…
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Positive semi-definite vs positive definite

I am confused about the difference between positive semi-definite and positive definite. May I understand that positive semi-definite means symmetric and $x'Ax \ge 0$, while positive definite means symmetric and $x'Ax \gt 0$?
pippp
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Is this matrix obviously positive definite?

Consider the matrix $A$ whose elements are $A_{ij} = a^{|i-j|}$ for $-1
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Are positive definite matrices robust to "small changes"?

Let $A$ be a positive-definite matrix and let $B$ be some other symmetric matrix. Consider the matrix $$ C=A+\varepsilon B. $$ for some $\varepsilon>0$. Is it true that for $\varepsilon$ small enough $C$ is also positive definite?
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