Why covariance matrix is positive definite

Hence the matrix has to be symmetric. You can always find a transformation of your variables in a way that the covariance-matrix becomes diagonal. On the diagonal, you find the variances of your transformed variables which are either zero or positive, it is easy to see that this makes the transformed matrix positive semidefinite.

However, since the definition of definity is transformation-invariant, it follows that the covariance-matrix is positive semidefinite in any chosen coordinate system. When you estimate your covariance matrix that is, when you calculate your sample covariance with the formula you stated above, it will obv. It also has to be positive semidefinite I think , because for each sample, the pdf that gives each sample point equal probability has the sample covariance as its covariance somebody please verify this , so everything stated above still applies.

Variance-Covariance matrices are always symmetric, as it can be proven from the actual equation to calculate each term of said matrix. Also, Variance-Covariance matrices are always square matrices of size n, where n is the number of variables in your experiment.

With PCA, you determine the eigenvalues of the matrix to see if you could reduce the number of variables used in your experiment. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Is a sample covariance matrix always symmetric and positive definite? Ask Question. Asked 8 years, 7 months ago. Active 2 years, 4 months ago.

Viewed 72k times. Currently my problem has a sample of observation vectors and 24 dimensions. Improve this question. Morten Morten 1 1 gold badge 9 9 silver badges 11 11 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. Zen Zen The necessary and sufficient conditions for it to be so are described in my comment to Konstantin's answer. Konstantin Schubert Konstantin Schubert 2 2 silver badges 12 12 bronze badges. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

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Create a free Team What is Teams? Learn more. What is the proof that covariance matrices are always semi-definite? Ask Question. Asked 9 years, 8 months ago. Active 2 years, 6 months ago. Viewed 72k times. Rodrigo de Azevedo It is my mistake. Add a comment. Active Oldest Votes.

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