Page 282 - 35Linear Algebra

282 Diagonalizing Symmetric Matrices

 
z 1
.
n
†
n
(c) Let x =  . ∈ C . Let x = z 1 · · · z n ∈ C (a 1 × n
 .
z n
†
complex matrix or a row vector). Compute x x. Using the result
†
of part 1a, what can you say about the number x x? (E.g., is it
real, imaginary, positive, negative, etc.)
T
(d) Suppose M = M is an n×n symmetric matrix with real entries.
Let λ be an eigenvalue of M with eigenvector x, so Mx = λx.
Compute:
†
x Mx
†
x x
T
(e) Suppose Λ is a 1 × 1 matrix. What is Λ ?
†
(f) What is the size of the matrix x Mx?
(g) For any matrix (or vector) N, we can compute N by applying
† T
complex conjugation to each entry of N. Compute (x ) . Then
T
†
compute (x Mx) . Note that for matrices AB + C = AB + C.
(h) Show that λ = λ. Using the result of a previous part of this
problem, what does this say about λ?

Hint

2. Let
 
a
b
x 1 =   ,
c
2
2
2
where a + b + c = 1. Find vectors x 2 and x 3 such that {x 1 , x 2 , x 3 }
3
is an orthonormal basis for R . What can you say about the matrix P
whose columns are the vectors x 1 , x 2 and x 3 that you found?
linear
3. Let V 3 v 6= 0 be a vector space, dimV = n and L : V −−−→ V .

n
2
(a) Explain why the list of vectors (v, Lv, L v, . . . , L v) is linearly
dependent.

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