2331_Notes_6o2_fill - Math 2331 Linear Algebra Section 6.2...

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

View Full Document Right Arrow Icon
Math 2331 Linear Algebra Section 6.2 Diagonalizing a Matrix When x is an eigenvector, multiplication by the matrix A is nothing more than multiplying x by a scalar. The interconnected matrix product simply translates into scalar multiplication. This has the wonderful effect of "separating" the affects of A into its multiple of each eigenvalue. This makes the transformation Ax SO MCUH EASIER to work with. It becomes like a diagonal matrix. Multiplying by a diagonal matrix simply multiples each component of the vector by the entry in the corresponding row. Eigenvectors that come from different eigenvalues are linearly independent. 2 x 2 proof -
Background image of page 1

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

View Full DocumentRight Arrow Icon
Let A be n x n with n distinct eigenvalues and let S be the matrix of the corresponding eieenvectors as its colums - Then AS = Let 1 2 0 ... 0 0 ... 0 0 0 ... 0 0 0 ... n λ ⎡⎤ ⎢⎥ Λ= ⎣⎦ Then S So, 1 1 AS S ASS SA S Example: 15 06 A ⎛⎞ = ⎜⎟ ⎝⎠
Background image of page 2
Notes 1. Any matrix that has no repeated eigenvalues can be diagonalized.
Background image of page 3

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

View Full DocumentRight Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

2331_Notes_6o2_fill - Math 2331 Linear Algebra Section 6.2...

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