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Week10 - LECTURE 1 LECTURE 2 0 Distinct eigenvalues I...

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LECTURE 1 LECTURE 2 0. Distinct eigenvalues I haven’t gotten around to stating the following important theorem: Theorem: A matrix with n distinct eigenvalues is diagonalizable. Proof (Sketch) Suppose n = 2, and let λ 1 and λ 2 be the eigenvalues, ~v 1 ,~v 2 the eigenvectors. But for all we know, ~v 1 and ~v 2 are not linearly independent! Suppose they’re not; then we have c 1 ~v 1 + c 2 ~v 2 = ~ 0 with c 1 and c 2 not both 0. Multiplying both sides by A , get c 1 λ 1 ~v 1 + c 2 λ 2 ~v 2 = ~ 0 . Multiplying first equation by λ 1 gives c 1 λ 1 ~v 1 + c 2 λ 1 ~v 2 = ~ 0 and subtracting gives c 2 ( λ 2 - λ 1 ) ~v 2 = ~ 0 and from this–since λ 2 6 = λ 1 !–we can conclude c 2 = 0. It follows similarly that c 1 = 0, contradiction. A similar argument gives the result for any n , but it’s not as easy as Strang makes it seem; it requires the fact that the Vandermonde matrix is invertible (see Strang, p.98). Apropos of nothing, I also want to comment: Fact. A is invertible if and only if 0 is not an eigenvalue of A . 1. Symmetric, Hermitian, unitary matrices Spectral theorem: A (real) symmetric matrix is diagonalizable. Strangely enough, the best way to prove this (and I think Strang’s proof is very good) is to use complex matrices. Definitions: Recall that the complex conjugate of a number a + bi is a - bi . Similarly, the complex conjugate of a matrix A is the matrix obtained by replacing each entry with its complex conjugate. A H (“A Hermitian”) is the complex conjugate of A T . 1
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A is symmetric if A = A T . A is Hermitian if A = A H . A is unitary if AA H = 1. Note that “unitary” is the complex analogue of “orthogonal.” Indeed, a real unitary matrix is orthogonal. Note also that ( AB ) H = B H A H . Give the example of heat diffusion on a circle to suggest the ubiquity of symmetric matrices. Examples: A typical Hermitian matrix is 1 i - i 1 . Compute, just for fun, that the eigenvalues are 0 and 2. That they’re real numbers, despite the fact that the matrix is complex, is no coincidence! We might want to analyze this before we think about unitary matrices too much. Definition: Let ~v be a vector with complex entries. We define k ~v k = ~v H ~v (positive square root.) Before we prove the spectral theorem, let’s prove a theorem that’s both stronger and weaker.
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