Math 223 Assignment 9
1. Find an orthonormal basis for R3 by applying Gram-Schmidt to the three vectors: 12
−2
 ,
 43
2
 ,
 12
1
 .
Please recall that we can change a vector with fractional entries to one with integral entries by
rescaling and we do not need to normalize until the last step.
2. Please note the very important fact that an n × n symmetric matrix always is diagonalizable
with an orthonormal basis of eigenvectors for Rn. This will be covered in class but perhaps not
completed before the due date of this assignment. Find orthonormal bases for of eigenvectors for
the following matrices :
A =
[
1 2
2 4
]
, B =
 0 10 1010 5 0
10 0 −5
 , C =
−1 −2 1−2 2 −2
1 −2 −1

Hint: for B, 0 is an eigenvalue, and for C, −2 is an eigenvalue.
3. Let A be a 3×3 matrix with det(A−λI) = −(λ3+aλ2+bλ+c). Show that if A is diagonalizable,
then the following equation is true
A3 + aA2 + bA+ cI = 0
(this equation is in fact true for any 3 × 3 matrix and is a special case of the Cayley-Hamilton
Theorem).
4. Let {u1,u2, . . . ,uk} be non-zero vectors satsifying ui · uj = 0 for all pairs i, j with i 6= j. Show
that {u1,u2, . . . ,uk} are linearly independent.
5. Let A be an n× n symmetric matrix with eigenvalues λ1, λ2, . . . , λn (some may repeat) and an
orthonormal basis of eigenvectors u1,u2, . . . ,un (Aui = λiui). Then show that
A =
n∑
i=1
λiuiu
T
i .
(thus A is a sum of n symmetric rank 1 matrices)
6. (from an exam) Consider a matrix
[
a b
c d
]
. We could attempt to solve for A−1 by letting
A−1 =
[
x y
z t
]
with four variables x, y, z, t and then AA−1 = I becomes a system of 4 equations in 4 unknowns
with an associated 4 × 4 matrix B. What is the rank of the 4 × 4 system of equations assuming
A−1 exists? Explain. Can you say anything about the rank of B if det(A) = 0? Explain.

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