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Karthiksrao
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I am curious to know how g'Bg compares with g'g when B is a positive definite matrix and g is a vector.
Is g'Bg >= g'g ?
Thanks,
Karthik
Is g'Bg >= g'g ?
Thanks,
Karthik
A positive definite matrix is a square matrix where all the eigenvalues are positive. This means that when we multiply the matrix by any non-zero vector, the resulting product is always positive.
Some properties of a positive definite matrix include: all eigenvalues are real and positive, all principal minors are positive, and it is nonsingular (meaning it has an inverse).
Positive definite matrices are used in many areas of mathematics, including optimization, statistics, and linear algebra. They are particularly useful in optimization problems, as they can help determine the minimum or maximum values of a function.
To determine if a matrix is positive definite, we can use several methods. One method is to check the signs of the eigenvalues – if they are all positive, the matrix is positive definite. Another method is to check if all the principal minors are positive.
No, a matrix cannot be both positive definite and positive semidefinite. A positive semidefinite matrix has all non-negative eigenvalues, whereas a positive definite matrix has all positive eigenvalues. Therefore, a positive definite matrix is a stricter condition than positive semidefinite.