Tests for Positive Definiteness of a Matrix

In order to perform Cholesky Decomposition of a matrix, the matrix has to be a positive definite matrix. I have listed down a few simple methods to test the positive definiteness of a matrix.

Methods to test Positive Definiteness:

Remember that the term positive definiteness is valid only for symmetric matrices.

Test method 1: Existence of all Positive Pivots

For a matrix to be positive definite, all the pivots of the matrix should be positive. Hmm.. What is a pivot ?

Pivots:

Pivots are the first non-zero element in each row of a matrix that is in Row-Echelon form. Row-Echelon form of a matrix is the final resultant matrix of Gaussian Elimination technique.

In the following matrices, pivots are encircled.

A positive definite matrix will have all positive pivots. Only the second matrix shown above is a positive definite matrix. Also, it is the only symmetric matrix.

Test method 2: Determinants of all upper-left sub-matrices are positive:

Determinant of all upper-left sub-matrices must be positive.

Break the matrix in to several sub matrices, by progressively taking upper-left elements. If the determinants of all the sub-matrices are positive, then the original matrix is positive definite.

Is the following matrix Positive Definite?

Find the determinants of all possible upper sub-matrices.

Test method 3: All Positive Eigen Values

If all the Eigen values of the symmetric matrix are positive, then it is a positive definite matrix.

Is if following matrix Positive definite ?

Since, not all the Eigen Values are positive, the above matrix is NOT a positive definite matrix.

There exist several methods to determine positive definiteness of a matrix. The method listed here are simple and can be done manually for smaller matrices.

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External resource:

1) Online tool to generate Eigen Values and Eigen Vectors↗

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See also:

[1]	An Introduction to Estimation Theory
[2]	Bias of an Estimator
[3]	Minimum Variance Unbiased Estimators (MVUE)
[4]	Maximum Likelihood Estimation
[5]	Maximum Likelihood Decoding
[6]	Probability and Random Process
[7]	Likelihood Function and Maximum Likelihood Estimation (MLE)
[8]	Score, Fisher Information and Estimator Sensitivity
[9]	Introduction to Cramer Rao Lower Bound (CRLB)
[10]	Cramer Rao Lower Bound for Scalar Parameter Estimation
[11]	Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)
[12]	Efficient Estimators and CRLB
[13]	Cramer Rao Lower Bound for Phase Estimation
[14]	Normalized CRLB - an alternate form of CRLB and its relation to estimator sensitivity
[15]	Cramer Rao Lower Bound (CRLB) for Vector Parameter Estimation
[16]	The Mean Square Error – Why do we use it for estimation problems
[17]	How to estimate unknown parameters using Ordinary Least Squares (OLS)
[18]	Essential Preliminary Matrix Algebra for Signal Processing
[19]	Why Cholesky Decomposition ? A sample case:
[20]	Tests for Positive Definiteness of a Matrix
[21]	Solving a Triangular Matrix using Forward & Backward Substitution
[22]	Cholesky Factorization - Matlab and Python
[23]	LTI system models for random signals – AR, MA and ARMA models
[24]	Comparing AR and ARMA model - minimization of squared error
[25]	Yule Walker Estimation
[26]	AutoCorrelation (Correlogram) and persistence – Time series analysis
[27]	Linear Models - Least Squares Estimator (LSE)
[28]	Best Linear Unbiased Estimator (BLUE)