Questions tagged [positive-semidefinite]

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

If $M$ is a positive semidefinite matrix then it has some additional properties which can be found in this Wikipedia article. You can also use this tag if one or more of these properties leads back to $M$ being positive-semidefinite.

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Intuitive explanation of a positive semidefinite matrix

What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors. The definition is: A $n$…
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Is the product of symmetric positive semidefinite matrices positive definite?

I see on Wikipedia that the product of two commuting symmetric positive definite matrices is also positive definite. Does the same result hold for the product of two positive semidefinite matrices? My proof of the positive definite case falls apart…
nullUser
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What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3}…
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Prove that every positive semidefinite matrix has nonnegative eigenvalues

There is a theorem which states that every positive semidefinite matrix only has eigenvalues $\ge0$ How can I prove this theorem?
kasper van lange
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Properties of the cone of positive semidefinite matrices

The set of positive semidefinite symmetric real matrices forms a cone. We can define an order over the set of matrices by saying $X \geq Y$ if and only if $X - Y$ is positive semidefinite. I suspect that this order does not have the lattice…
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How to check if a symmetric $4\times4$ matrix is positive semidefinite?

How does one check whether symmetric $4\times4$ matrix is positive semidefinite? What if this matrix has also rank deficiency: is it rank $3$?
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Under what circumstance will a covariance matrix be positive semi-definite rather than positive definite?

I have a covariance matrix: $\operatorname{cov}(\mathbf{X}, \mathbf{X}) = \operatorname{E}[(\mathbf{X} - \operatorname{E}[\mathbf{X}])(\mathbf{X} - \operatorname{E}[\mathbf{X}])^T]$ According to Wikipedia, it should be a positive semi-definite…
Damien
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Why do mathematicians use only symmetric matrices when they want positive semi-definite matrices?

Why do mathematicians use only symmetric matrices when they want positive semi/definite matrices? I mean I haven't seen using non-symmetric positive semi/definite matrices. If non-symmetric positive semi/definite matrices exist can those be always…
triomphe
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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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Is $U=V$ in the SVD of a symmetric positive semidefinite matrix?

Consider the SVD of matrix $A$: $$A = U \Sigma V^\top$$ If $A$ is a symmetric, positive semidefinite real matrix, is there a guarantee that $U = V$? Second question (out of curiosity): what is the minimum necessary condition for $U = V$?
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Does a positive semidefinite matrix always have a non-negative trace?

If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace. If not, any counter-example? Thanks.
MIMIGA
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Decomposition of a positive semidefinite matrix

Let $Y \in \mathbb{R}^{n \times n}$ be a symmetric, positive semidefinite matrix such that $Y_{kk} = 1$ for all $k$. This matrix is supposed to be factorized as $Y = V^T V$, where $V \in \mathbb{R}^{n \times n}$. Does this…
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Is the trace of the product of two positive semidefinite matrices always nonnegative?

Is $\mbox{tr}(XY) \geq 0$ for all $X, Y \in \Bbb S_+$?
costa
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Meaning of $x^T A x$

I've seen the term $x^T A x$, where $A$ is a square and usually symmetric matrix, come up in a bunch of different areas of linear algebra. Places I've seen it include defining the Raleigh quotient, defining positive/negative semi-definite matrices,…
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Is there any intuition why the following matrix is positive semidefinite?

I have the following 8 by 8 square matrix, which is positive semidefinite: \begin{bmatrix}3&1&1&-1&1&-1&-1&-3\\1&3&-1&1&-1&1&-3&-1& \\ 1&-1&3&1&-1&-3&1&-1 \\ -1&1&1&3&-3&-1&-1&1 \\ 1&-1&-1&-3&3&1&1&-1 \\ -1&1&-3&-1&1&3&-1&1 \\ -1&-3&1&-1&1&-1&3&1 …
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