1.Positive Definite Matrix
LetA be an n-dimensional (real) symmetric matrix. If for any non-zero vectorz, it holds thatz(T)Az > 0, wherez(T) denotes the transpose ofz, thenA is called a positive definite matrix.
2.Methods to Determine Positive Definiteness of a Matrix
a.Calculate all eigenvalues of the symmetric matrixA. If all eigenvalues ofA are positive, thenA is positive definite;
b.Calculate all leading principal minors of the symmetric matrixA. If all leading principal minors ofA are greater than zero, thenA is positive definite.
3.Understanding Functions
Eigenvalue Calculationeig
Matrix Rankrank
4.Programming Example
(1) Calculate the rank of the matrix
A= [1 0 1 0
2 0 2 0 ]
Program:
A= [1 0 1 0
2 0 2 0 ];
rank(A)
Execution Result:
ans =
1
(2) Determine the positive definiteness of the following matrixA.
A=[10 12 10
12 2 15
10 15 4]
Program:
A=[10 12 10
12 2 15
10 15 4];
eig(A)
Execution Result:
ans =
-12.2928
-1.7697
30.0625
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