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vaibhavtewari
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Hi, i was wondering is there a way I can find the eigenvalues of a positive definite matrix. ?
vaibhavtewari said:What are your thoughts regarding QR decomposition ?
A positive definite matrix is a square matrix where all of its eigenvalues are positive. In other words, when the matrix is multiplied by any non-zero vector, the result is always a positive value.
Eigenvalues are scalar values that represent how a linear transformation affects a vector. Eigenvectors are non-zero vectors that, when multiplied by a matrix, result in a scalar multiple of itself, i.e. the vector's direction remains unchanged.
The eigenvalues of a positive definite matrix can be found by solving the characteristic equation, which is det(A - λI) = 0, where A is the matrix and λ is the eigenvalue. The solutions to this equation are the eigenvalues of the matrix.
Positive eigenvalues in a positive definite matrix indicate that the matrix is a local minimum, meaning that the matrix has a smaller value at its center than at any other nearby point. This is useful in optimization problems, where we want to find the minimum value of a function.
Positive definite matrices have many applications in various fields such as physics, engineering, and computer science. They are used in optimization problems, machine learning algorithms, and in the analysis of physical systems such as quantum mechanics. They are also used in solving systems of linear equations and in diagonalizing matrices.