symmetric_matrices_2009_10_05_02

symmetric_matrices_2009_10_05_02 - 6 1 Symmetric and...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 6 - 1 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 6. Symmetric and Positive Matrices • Eigenvalues • Eigenvectors • Example: mechanical systems • Quadratic forms • Positive-definite matrices • Matrix square-roots • Example: mechanical systems • Ellipsoids • Example: navigation 6 - 2 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 Eigenvalues of Symmetric Matrices a matrix A ∈ R n × n is called symmetric if A = A T eigenvalues if A is symmetric, then the eigenvalues of A are real to show this, suppose x is an eigenvector of A with eigenvalue λ ∈ C . Then Ax = λx and x negationslash = 0 we will show that λ = ¯ λ . we know that ¯ x T Ax = λ ¯ x T x = λ n summationdisplay i =1 | x i | 2 also ¯ x T Ax = ( Ax ) T x = ¯ λ ¯ x T x = ¯ λ n summationdisplay i =1 | x i | 2 since bardbl x bardbl negationslash = 0 , we have λ = ¯ λ 6 - 3 Symmetric and Positive Matrices S. Lall, Stanford 2009.10.05.02 Eigenvectors of Symmetric Matrices for symmetric matrices: eigenvectors corresponding to distinct eigenvalues are orthogonal suppose Ax 1 = λ 1 x 1 and Ax 2 = λ 2 x 2 . Then x T 1 Ax 2 = ( Ax 1 ) T x 2 = λ 1 x T 1 x 2 and also x T 1 Ax 2 = λ 2 x T 1 x 2 therefore ( λ 1 − λ 2 ) x T 1 x 2 = 0 and since λ 1 negationslash = λ 2 we must have x T 1 x 2 = 0 • if A is symmetric and has n distinct eigenvalues, then its eigenvectors form an or- thogonal basis for the vector space R n 6 - 4...
View Full Document

{[ snackBarMessage ]}

Page1 / 12

symmetric_matrices_2009_10_05_02 - 6 1 Symmetric and...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online