# [R] matrix which results singular but at the same time positive definite

Fox, John jfox at mcmaster.ca
Thu Dec 10 14:41:44 CET 2015

```Dear Stefano,

You've already had a couple of informative responses directly addressing your question, but are you aware how ill-conditioned the matrix is (one of the responses alluded to this)?

> kappa(X, exact=TRUE)
[1] 7.313338e+12

> eigen(X)\$values
[1] 4.964711e+00 9.356881e-01 4.863392e-12 6.788344e-13

Two of the variables have variances around 10^0 and the other two around 10^-12. Of course, you haven't said anything about the context, but does it really make sense to analyze the data on these scales?

Best,
John

-----------------------------
John Fox, Professor
McMaster University
Hamilton, Ontario
Web: socserv.mcmaster.ca/jfox

> -----Original Message-----
> From: R-help [mailto:r-help-bounces at r-project.org] On Behalf Of Stefano Sofia
> Sent: December 10, 2015 5:08 AM
> To: r-help at r-project.org
> Subject: [R] matrix which results singular but at the same time positive definite
>
> Dear list users,
> through the "matrixcalc" package I am performing some checks of variance
> matrices (which must be positive definite).
> In this example, it happens that the matrix A here reported is singular but
> positive definite. Is it possible?
>
>               [,1]          [,2]          [,3]          [,4]
> [1,]  1.904255e-12 -1.904255e-12 -8.238960e-13 -1.240294e-12 [2,] -
> 1.904255e-12  3.637979e-12  1.364242e-12  1.818989e-12 [3,] -8.238960e-13
> 1.364242e-12  4.809988e+00  7.742369e-01 [4,] -1.240294e-12  1.818989e-12
> 7.742369e-01  1.090411e+00
>
> print(is.non.singular.matrix(A, tol = 1e-18)) FALSE print(is.positive.definite(A,
> tol=1e-18)) TRUE
>
> Is there something wrong with this matrix?
> Any comment will be appreciated.
> Stefano
>
>
> ________________________________
>
> AVVISO IMPORTANTE: Questo messaggio di posta elettronica può contenere
> informazioni confidenziali, pertanto è destinato solo a persone autorizzate alla
> ricezione. I messaggi di posta elettronica per i client di Regione Marche
> possono contenere informazioni confidenziali e con privilegi legali. Se non si è il
> destinatario specificato, non leggere, copiare, inoltrare o archiviare questo
> messaggio. Se si è ricevuto questo messaggio per errore, inoltrarlo al mittente
> ed eliminarlo completamente dal sistema del proprio computer. Ai sensi
> dell’art. 6 della DGR n. 1394/2008 si segnala che, in caso di necessità ed
> urgenza, la risposta al presente messaggio di posta elettronica può essere
> visionata da persone estranee al destinatario.
> IMPORTANT NOTICE: This e-mail message is intended to be received only by
> persons entitled to receive the confidential information it may contain. E-mail
> messages to clients of Regione Marche may contain information that is
> confidential and legally privileged. Please do not read, copy, forward, or store
> this message unless you are an intended recipient of it. If you have received
> this message in error, please forward it to the sender and delete it completely