# Diagonal matrix

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For the problem, refer to the linear transformation T: R^3→R^3 given by T(x) = T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z).

The matrix A = [T]_E is similar to a diagonal matrix D = [T]_F. Write the diagonal matrix D, and demonstrate that it is indeed similar to A by producing the appropriate non-singular matrix and its inverse.

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#### Solution Preview

T(x, y, z) = (2x + 2z, x - y + z, 2x + 2z)

T(1, 0, 0)= (2, 1, 2)

T(0, 1, 0)= (0, -1, 0)

T(0, 0, 1)= (2, 1, 2)

Then the matrix is:

[2 0 2]

A= [1 -1 1]

...

#### Solution Summary

This shows how to write diagonal matrix D, and demonstrate that it is similar to another matrix.

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