MATHS

MATHS
1. Find the spectra and eigenvetors for the two matrices below. Show your work.
A =
?
?
3 5 3
0 4 6
0 0 1
?
? A =
?
?
a 1 0
1 a 1
0 1 a
?
?
2. Find the matrix A for each of the indicated linear transformation y = Ax. Find its eigenvalues
and eigenvectors.
(a) Reflection about the x-axis in R2
. Here x = [x y].
(b) Orthogonal projection of R3 onto the plane x = y. Here x = [x y z].
3. Prove that trace of a square real or complex matrix A equals the sum of its eigenvalues. This
fact is often a useful check on the accuracy of eigenvalue calculations. Demonstrate with an
example of your choosing.
4. Prove that the eigenvectors of a real symmetric matrix corresponding to different eigenvalues
are orthogonal.
5. Do there exist real symmetric 3 × 3 matrices that are orthogonal (except for the unit matrix
I)?
6. Prove that Hermitian, skew-Hermitian and unitary matrices are all normal matrices.
7. Find the similarity transformation that diagonalizes the following matrix. Show details of your
work.
A =
?
?
16 0 0
48 -8 0
84 -24 4
?
?
8. Use the power method to find the largest eigenvalue to 5 significant figures of the first matrix

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