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2331_Notes_6o2_fill

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

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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 -

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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 ⎛⎞ = ⎜⎟ ⎝⎠
Notes 1. Any matrix that has no repeated eigenvalues can be diagonalized.

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

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