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.
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.
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.
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multiplying (square) matrices is complicated, we have n^{2} inner products of rows and columns to consider, which is:
n^{3} + n arithmetical operations in all (n products in each inner product, plus a summation, times n^{2}).
multiplying diagonal matrices is much simpler, the resulting product is ALSO diagonal, and requires only n operations:
diag{a_{1},...,a_{n}}*diag{b_{1},...,b_{n}} = diag{a_{1}b_{1},...,a_{n}b_{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.