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Diagonalization Theorems
Theorem 3 (Diagonalization Theorem) (a) An m × m matrix A is diagonable if and only if A has m linearly independent eigenvectors. of A. (b) Suppose ~v 1 ,... , ~vm ∈ Cm^ is a linearly inde- pendent set of eigenvectors of A with corresponding eigenvalues λ 1 ,... , λm (so A~vi = λi~vi, for 1 ≤ i ≤ m). Then the matrix P = (~v 1 |... |~vm) is invertible and we have P −^1 AP = Diag(λ 1 ,... , λm).
Remark: Write
chA(t) = (t − λ 1 )m^1 · · · (t − λr)mr, where the λi’s are distinct, and let Bi be a basis of the corresponding eigenspaces EA(λi), 1 ≤ i ≤ r. Then it can be shown that B = B 1 ∪ B 2 ∪... ∪ Br is a linearly independent set. Thus: A is diagonable ⇔ B is a basis of Cm^ ⇔ #B = m ⇔ dim EA(λ 1 ) +... + dim EA(λr) = m.
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Corollary: If A has m distinct eigenvalues (i.e. if chA(t) has m distinct roots), then A is diagonable.
Warning: The converse of this corollary is false: a matrix A can be diagonable yet have repeated eigenvalues. Example: the m × m identity matrix I is diago- nal (hence diagonable), but has only one eigenvalue λ 1 = 1 (repeated m times).
Remark: If A is a real matrix (i.e. all entries of A are real numbers) and is symmetric, i.e. At^ = A, then A is automatically diagonable, as the the following useful result shows:
Theorem 4 (Principal Axis Theorem) If A is a real symmetric matrix, then A is orthog- onally diagonable; in other words, there exists an orthogonal matrix P (i.e. a real matrix satisfying P −^1 = P t) such that P −^1 AP is a diagonal matrix.
Remark: The name of this theorem comes from the fact that this theorem can used to show that quadrics in Rn^ centered at the origin (e.g. ellipses in R^2 , el- lipsoids in R^3 , etc.) can be rotated so that their principal axes are along the coordinate axes of Rn. 2
