If [itex]B=A^TA[/itex], write down what [itex]B_{ij}[/itex] and [itex]B_{ji}[/itex] are in terms of the elements of A and show that they're equal.rainwyz0706 said:For the first problem, I'm thinking of proving that AtA is symmetric, but I'm not sure which properties to use.
The diagonalisability problem is a mathematical problem that involves determining whether a given matrix can be transformed into a diagonal matrix through a change of basis. This problem is important in linear algebra and has applications in fields such as quantum mechanics and differential equations.
A matrix is diagonalisable if it can be transformed into a diagonal matrix through a change of basis. This can be determined by finding the eigenvalues and eigenvectors of the matrix and checking if the eigenvectors form a basis for the vector space. If they do, then the matrix is diagonalisable.
Diagonalisability has various applications in science, particularly in fields such as quantum mechanics and differential equations. In quantum mechanics, diagonalisability is used to find the energy levels of a system. In differential equations, diagonalisability is used to solve systems of linear differential equations.
Other problems related to diagonalisability include finding the inverse of a matrix, computing powers of a matrix, and solving systems of linear equations. These problems are all linked to diagonalisability as they involve matrix transformations and the use of eigenvectors and eigenvalues.
Diagonalisability has practical applications in fields such as engineering, physics, and computer science. For example, in engineering, diagonalisability is used in control systems to model and analyze the behavior of dynamic systems. In physics, it is used to understand the behavior of quantum systems. In computer science, it is used in data compression and signal processing.