How can I prove that a diagonal dominant matrix is a P-matrix? And how can I prove that if it is diagonally dominant for both rows and columns, it is also positive definite?
How can I prove that a diagonal dominant matrix is a P-matrix?
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linear-algebra
matrices
positive-definite
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0What is a P-matrix? Diagonally dominant is standard terminology, but P-matrix is not. – 2017-02-28
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0A real square matrix is said to be P-matrix if all its principal minors are positive. – 2017-02-28
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0I think you'll need to assume that the entries are positive real numbers. Clearly $A=\begin{bmatrix} -1 & 0\\ 0 & -1\end{bmatrix}$ is diagonally dominant but its principal minors are not all positive. – 2017-02-28
1 Answers
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Hint: I assume we're referring to a matrix with real entries.
- For part 1, use the Gershgorin circle theorem. Note that the determinant of a matrix is the product of its eigenvalues. Note that one should handle the possibility of complex eigenvalues (which necessarily come in conjugate pairs).
- For part 2: note that a matrix $A$ is positive definite if and only if the symmetric matrix $A + A^T$ is positive definite. Sylvester's criterion tells us that a symmetric $P$-matrix is necessarily positive definite. Now: how do we know that $A + A^T$ is a $P$-matrix?