Diagonalisation and Eigenbasis

In class on Wednesday I mentioned the following result.

A matrix is diagonalisable if and only if it possesses an eigenbasis, that is, a basis consisting of eigenvectors.

Here is the proof:

Suppose a matrix A (an n by n matrix) admits an eigenbasis. That is, there exists a basis of n eigenvectors for A, call them v_1, \ldots, v_n such that Av_i = \lambda_i v_i for all i. Form a matrix whose columns are the v_i (that is S = (v_1 \cdots v_n)). Then S is a change of basis matrix, and moreover, we have

S^{-1}ASe_i = S^{-1}Av_i = \lambda_i S^{-1} v_i = \lambda_i e_i

which shows that S^{-1}AS=D a diagonal matrix, with diagonal entries the corresponding eigenvalues of A.

Conversely, suppose that A is diagonalizable. ThenĀ there exists a change of basis matrix S such that AS = SD with D diagonal. Now this equation tells us that the n by n matrix AS is equal to the n by n matrix SD. In particular, each row of these matrices must agree. The i-th row of this equation is

Av_i = \lambda v_i

which tells us that the matrix S is full of eigenvectors. In particular, the eigenvectors of S form a basis and are thus an eigenbasis.

Here are some questions to think about. We can talk about these in office hours on Monday, or next class. [A hint for 1 and 2 is (as always!) the rank-nullity theorem!]

  1. In the first part of the proof, we built S with eigenvectors of A. Why could we be sure that S was invertible?
  2. In the second part of the proof, we concluded by showing that S was full of eigenvectors. Why does it follow that these vectors form a basis?
  3. Find the eigenvectors for the following matrices, and determine the sum of the dimensions of the eigenspaces [to save you some time – the eigenvalues for both are -2, -2, -3]:
  • A=\left(\begin{array}{ccc}0 & -6 & -4 \\5 & -11 & -6 \\-6 & 9 & 4\end{array}\right)
  • B = \left(\begin{array}{ccc}4 & 8 & -2 \\-3 & -6 & 1 \\9 & 12 & -5\end{array}\right)

NOTE: These questions are not going to be graded and are in no way part of your grade. They are just questions I think could be helpful to try and understand some of these new ideas.

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