MHB Why do we need to convert to a diagonal matrix?

[matqkks]

Apart from simplifying matrix powers, why do we want to diagonalize a matrix? Do they have any appealing application which can be used to motivate to study diagonal matrices.
Thanks for any answers.

 

[Ackbach]

Simplifying matrix powers IS the main application for diagonalization. Why? Because of the very general ODE $\dot{\mathbf{x}}=A\mathbf{x}$ for constant $A$. If $A$ is diagonalizable, then the solution $\mathbf{x}=e^{At}\mathbf{x}_{0}$ makes sense only if you can exponentiate the $At$. To do that, you can form the Taylor series using matrices. Then, to compute that Taylor series, the computations are much more tractable with a diagonal matrix.

 

[Deveno]

multiplying (square) matrices is complicated, we have n² inner products of rows and columns to consider, which is:

n³ + n arithmetical operations in all (n products in each inner product, plus a summation, times n²).

multiplying diagonal matrices is much simpler, the resulting product is ALSO diagonal, and requires only n operations:

diag{a₁,...,a_n}*diag{b₁,...,b_n} = diag{a₁b₁,...,a_nb_n}

even when n is small (like say n = 4), this is a tremendous savings of calculational effort (we only have 4 steps of arithmetic, rather than 68).

it also making calculating the determinant MUCH more tractable: the determinant is invariant under a similarity transform. for an nxn matrix, normally calculating it requires computing n! n-fold products and then summing these, whereas computing the determinant of a diagonal matrix requires just computing ONE n-fold product.

for example, computing a 5x5 determinant requires 121 arithmetical operations (even determining which 120 5-fold products to compute is tedious), whereas computing a 5x5 diagonal matrix's determinant can often be done in your head.

morevoer, if A is diagonalizable, diagonalizing A illustrates a deep connection between the diagonalized matrix and the eigenvalues of A, and the diagonalizing matrix P and the eigenvectors of A (and since P is invertible, that the eigenvectors form an eigenbasis).

the "catch" here is that not all matrices ARE diagonalizable. it turns out, however, that we can at least "semi-diagonalize" A into the sum:

D + N, where D is diagonal, and N is nilpotent.

this shows how important understanding nilpotent linear transformations is to "getting a good picture of bad matrices" (the diagonalizable ones being "good matrices").

if a matrix function can be represented as a power series (such as in the exponential example Ackbach gives), then computing the matrix function becomes a LOT easier if our matrix is diagonalizable.

unfortunately, the set of diagonalizable matrices isn't closed under matrix addition, which is a darn shame.