文档介绍：CHAPTER 7 Symmetric Matrices and Quadratic Forms
Chapter 7 Symmetric Matrices and Quadratic Forms
§ Diagonalization of Symmetric Matrices
§ Quadratic Forms
§ Diagonalization of Symmetric Matrices
What is Symmetric Matrices?
A symmetric matrix is a matrix A such that AT = A.
Example :
Symmetric:
Nonsymmetric:
§ Diagonalization of Symmetric Matrices
Properties of Symmetric Matrices
Such a matrix is necessarily square.
Its main diagonal entries are arbitrary.
Its other entries occur in pairs- on opposite sides of the main diagonal.
Review : the diagonalization process.
Example :If possible, diagonalize the matrix
Solution: The characteristic equation of A is
Standard calculations produce a basis for each eigenspace:
Normailize v1, v2 and v3 to produce the unit eigenvectors
Let
Then A = PDP-1. Since P is square and has orthonormal columns, P is an orthogonal matrix, and P-1 is simply PT.
§ Diagonalization of Symmetric Matrices
Theorem 1
If A is symmetric, then any two eigenvectors from different eigenspaces are orthogonal.
Proof: Let v1 and v2 be eigenvectors that correspond to
distinct eigenvalues, say λ1 and λ2. To show that v1·v2 = 0,
compute
∴
∵
∴
A matrix A is said to be orthogonally diagonalizable if there are an orthogonal matrix P (with P-1 = PT) and a diagonal matrix D such that
To orthogonally diagonalize an n×n matrix, we must find n linearly independent and orthonormal eigenvectors. If A is orthogonally diagonalizable as in (1), then
§ Diagonalization of Symmetric Matrices
Theorem 2 shows that, every symmetric matrix is orthogonally diagonalizable.
Theorem 2
An n×n matrix A is orthogonally diagonalizable if and only if A is symmetric matrix.
Example:Orthogonally diagonalize the matrix
Its characteristic equation is
Solution: Compute to produce bases for the eigenspaces:
Although v1 and v2 are linearly independent, they
are not orthogonal.