13 Math 2362 – Symmetric matrices and diagonalization (7.2)
MATH 2362 – Symmetric matrices and diagonalization (7.2)
Example (last time): Diagonalize the symmetric matrix
A=⎡⎢⎣211121112⎤⎥⎦
Characteristic polynominal: f(λ)=(λ−1)2(λ−4)
Eigenvectors: λ=1––––––:→v1=⎡⎢⎣−110⎤⎥⎦,→v2=⎡⎢⎣−101⎤⎥⎦ (and their non-zero linear combinations)
λ=4––––––:→v3=⎡⎢⎣111⎤⎥⎦ (and its no-zero multiples)
Then P=[→v1→v2→v3]=⎡⎢⎣−1−11101011⎤⎥⎦ diagonalizes A, with P−1AP=D=⎡⎢⎣100010004⎤⎥⎦
Theorem: If A is symmetric, any two eigenvectors associated with distinct eigenvalues are orthogonal.
Definition: A square matrix P is called orthogonal is P is invertible and P−1=PT.
Theorem: A matrix P is orthogonal if and only if the columns (and rows) of P form an orthonormal set.
MATH 2362 – Orthogonal Diagonalization (7.2)
Defintion: A square matrix A is called orthogonally diagonalizable if there is an orthogonal matrix diagonalizing A, i.e. there is an orthogonal P such that P−1AP=D.
Theorem: An n x n matrix A is orthogonally diagonalizable if and only if A is symmetric.
Proof:
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Idea of proof: if A is symmetric matrix, all its eigenspaces are full-dimensional (i.e. geometric multiplicity=algebraic multiplicity for every eigenvalue).
⟹
Corollary: Symmetric matrices are always diagonalizable.
Example: Orthogonally diagonalize the symmetric matrix A=⎡⎢⎣211121112⎤⎥⎦
Characteristic polynominal: f(λ)=(λ−1)2(λ−4)
Eigenvectors:
λ=1––––––:→v1=⎡⎢⎣−110⎤⎥⎦,→v2=⎡⎢⎣−101⎤⎥⎦ (and their non-zero linear combinations)
λ=4––––––:→v3=⎡⎢⎣111⎤⎥⎦ (and its no-zero multiples)
Then P=[→v1→v2→v3]=⎡⎢⎣−1−11101011⎤⎥⎦ diagonalizes A, with P−1AP=D=⎡⎢⎣100010004⎤⎥⎦