Problem 5
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The eigenvalues of a $4 \times 4$ matrix, $A$, are 10, 4, 4, and 1.
Execute Gram-Schmidt process on the
following two vectors to find an orthonormal
basis set: $v_1 = (2, 2)$, $v_2 = (10, \dots)$. After normalization,
we have two new vectors $w_1 = (x_1, x_2)$ and $w_2 = (x_2, x_2)$.
What is the trace, i.e. sum of the diagonal
entries, of the matrix $A$?
What is the second largest eigenvalue of $A^2$?
What is the largest eigenvalue of $A^{-1}$?
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Problem 6
1.0 point possible (graded, results hidden)
Imagine you have a $3 \times 3$ matrix that
transforms every point into a mirror
opposite across a plane $P$. All points on $P$
remain as they are, and all points outside $P$
are transformed to a point on the exact
opposite side of $P$, using our accustomed
notion of reflection. Then,
how many eigenvalues does this 3D reflection
matrix have?
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