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MA 106: Linear Algebra Tutorial 5: Prof. B.V. Limaye IIT Dharwad

This document is a tutorial for a linear algebra course that covers the following topics: 1. Orthonormalizing ordered subsets of vectors. 2. Using the Gram-Schmidt process to orthonormalize an ordered subset and express another vector as a linear combination of the resulting basis vectors. 3. Showing that the rows of a unitary matrix form an orthonormal subset.

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116 views4 pages

MA 106: Linear Algebra Tutorial 5: Prof. B.V. Limaye IIT Dharwad

This document is a tutorial for a linear algebra course that covers the following topics: 1. Orthonormalizing ordered subsets of vectors. 2. Using the Gram-Schmidt process to orthonormalize an ordered subset and express another vector as a linear combination of the resulting basis vectors. 3. Showing that the rows of a unitary matrix form an orthonormal subset.

Uploaded by

amar Baronia
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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MA 106: Linear Algebra

Tutorial 5

Prof. B.V. Limaye


IIT Dharwad

Friday, 9 February 2018

B.V. Limaye, IITDH MA 106: Tut-05


Tutorial 5 (on Lectures 13, 14, and 15)

1. Orthonormalize the following ordered subsets of K4×1 .


(i) (e1 , e1 + e2 , e1 + e2 + e3 , e1 + e2 + e3 + e4 )
(ii) e1 + e2 + e3 + e4 , −e1 + e2 , −e1 + e3 , −e1 + e4 ).
2. Use
( [ the G-S OP]Tto [orthonormalize ]T [the ordered ]subset
T)
1 −1 2 0 , 1 1 2 0 , 3 0 0 1 and
obtain an ordered orthonormal set (u1 , u2 , u3 ). Also, find
u4 such that (u1 , u2 , u3 , u4 ) is an orthonormal basis for
[ ]T
K4×1 . Express the vector 1 −1 1 −1 as linear
combination of these four basis vectors.
3. Show that A ∈ Kn×n is unitary if and only if its rows form
an orthonormal subset of K1×n .

B.V. Limaye, IITDH MA 106: Tut-05


4. Let E := (e1 , . . . , en ) be the standard basis for Kn×1 , and
let F := (u1 , . . . , un ) be an orthonormal basis for Kn×1 . If
I denotes the identity map from Kn×1 to Kn×1 , then
show that the matrix MFE (I ) is unitary.
5. Let A ∈ Cn×n and let λ be an eigenvalue of A. Show that
p(λ) is an eigenvalue of p(A) for every polynomial p(t).
6. Suppose A ∈ C3×3 satisfies A3 − 6A2 + 11A = 6I.
If 5 ≤ det A ≤ 7, determine the eigenvalues of A.
Is A diagonalizable?
7. Let A ∈ Kn×n , and let λ1 , . . . , λn be the eigenvalues of A
with a corresponding orthonormal set of eigenvectors
u1 , . . . , un . Show that A = λ1 u1 u∗1 + · · · + λn un u∗n .
(xy∗= outer product of x, y)

B.V. Limaye, IITDH MA 106: Tut-05


8. Let A ∈ Kn×n , and λ ∈ K.
(i) Show that λ is an eigenvalue of A if and only λ is an
eigenvalue of A∗ .
(ii) Let A be normal. Show that ∥Ax∥ = ∥A∗ x∥ for all
x ∈ Kn×1 and deduce that x is an eigenvecor of A
corresponding to λ if and only if x is an eigenvector of A∗
correspnding to λ. Further, show that if µ ̸= λ,
x ∈ N (A − λI) and y ∈ N (A − µI), then x ⊥ y.
(iii) Let A be unitary. Show that ∥Ux∥ = ∥x∥ for all
x ∈ Kn×1 . If λ is an eigenvalue of A, show that |λ| = 1.
(iv) Let K = C and A skew self-adjoint, and let λ be an
eigenvalue of A. Show that iλ ∈ R.
9. Let A ∈ Cn×n , and let λ1 , . . . , λn be the eigenvalues of A
counting algebraic multiplicities.
∑ If A = [a∑
jk ], then show
that A is normal ⇐⇒ 1≤j,k≤n |ajk | =
2
j=1 |λj | .
n 2

B.V. Limaye, IITDH MA 106: Tut-05

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