The above process has been quoted as a simple application of eigenvalues and eigenvectors in my text book.
Could anyone explain briefly (I'm new to further matrices), why it is desirable to be able to diagonalise a matrix? What use is it in further mathematics, scientific/engineering problems?
Thanks in advance.
Could anyone explain briefly (I'm new to further matrices), why it is desirable to be able to diagonalise a matrix? What use is it in further mathematics, scientific/engineering problems?
Thanks in advance.
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Probably the most significant application of matrix diagonalization would be in relation to differential equations. A linear system of differential equations can be always be framed as a matrix differential equation. Even when analyzing a nonlinear system, one can often consider a linear stability analysis giving rise to a linear system.
If you think about a simple scalar problem dy/dx = ay, we know that the solution is y = Ce^(ax). Suppose that x(t) is a vector valued function in IRⁿ and A is an n x n matrix. One can consider the system
dx/dt = Ax.
In comparison to the scalar case, one might propose that the solutions should look like
x(t) = x_0e^(At)
where x_0 is a constant vector. Of course you have to define what you mean by e^(At) where A is a matrix. It turns out that you can define this and it is done exactly the way you'd think it should be done in terms of series. In practice, the matrix exponential can be computed not by a series but by appealing to diagonalization.
Diagonalization separates the action of the matrix (in terms of eigenvalues) into the elementary basis. All variables are completely decoupled making it super simple to solve an algebraic or diff. eq. problem. You use the transformation matrix to diagonalize. Solve the really easy diagonal problem and then transform back.
Not all matrices are diagonalizable, but there is a generalization of a diagonal matrix called a Jordan matrix. You can google Jordan form to see more about that.
If you think about a simple scalar problem dy/dx = ay, we know that the solution is y = Ce^(ax). Suppose that x(t) is a vector valued function in IRⁿ and A is an n x n matrix. One can consider the system
dx/dt = Ax.
In comparison to the scalar case, one might propose that the solutions should look like
x(t) = x_0e^(At)
where x_0 is a constant vector. Of course you have to define what you mean by e^(At) where A is a matrix. It turns out that you can define this and it is done exactly the way you'd think it should be done in terms of series. In practice, the matrix exponential can be computed not by a series but by appealing to diagonalization.
Diagonalization separates the action of the matrix (in terms of eigenvalues) into the elementary basis. All variables are completely decoupled making it super simple to solve an algebraic or diff. eq. problem. You use the transformation matrix to diagonalize. Solve the really easy diagonal problem and then transform back.
Not all matrices are diagonalizable, but there is a generalization of a diagonal matrix called a Jordan matrix. You can google Jordan form to see more about that.