A few notes on Matrix Algebra
Raghuveer Parthasarathy
Department of Physics, University of Oregon
for Physics 351: Foundations of Physics II
(Dated: November 5, 2007)
We’ll very briefly explain a few aspects of linear alge-
bra. This discussion is sufficient for the problems encoun-
tered in your homework assignments. To learn more, see
any linear algebra textbook, e.g.
Howard Anton, Elementary Linear Algebra (7th Ed.),
John Wiley and Sons, New York, 1994.
I. MATRICES
A matrix is a rectangular array of numbers. The ma-
trix A, below, is a 3 ×3 matrix, with elements aij, where
the indices iand jrun from 1 to 3:
A=
a11 a12 a13
a21 a22 a23
a31 a32 a33
.(1)
The matrix B, below, is a 3 ×1 matrix:
B=
b11
b21
b31
.(2)
The product AB is C, also a 3×1 matrix. The elements
ci1of Care given by
ci1=
3
X
j=1
aijbj1,
where Σ indicates a sum. In other words, each element of
Cis the sum of the products of a row of Awith a column
of B. (Here, Bonly has one column.)
C=AB =
a11b11 +a12b21 +a13 b31
a21b11 +a22b21 +a23 b31
a31b11 +a32b21 +a33 b31
.(3)
This is shorthand for three linear equations:
a11b11 +a12b21 +a13 b31 =c11 (4)
a21b11 +a22b21 +a23 b31 =c21 (5)
a31b11 +a32b21 +a33 b31 =c31.(6)
When dealing with Ncoupled oscillators, we’ll have
matrix equations that look like AB = 0, where 0 is a
N×1 array of zeros. The elements of Awill involve the
oscillation frequencies (ω), and the elements of Bwill
involve the amplitudes. We’ll want to find the condi-
tions such that the equation has a solution other than
the “trivial” one, B= 0. A fundamental theorem of lin-
ear algebra states that a non-trivial solution exists if and
only if the determinant of Ais zero. In the next section,
we’ll look at how to calculate a determinant.
Exercises. (Don’t turn these in.)
1. Let
A=
136
3−2 2
215
.(7)
B=
2
−1
0
.(8)
Calculate C=AB.
Answer:
C=
−1
8
3
.(9)
2. Write the following system of equations as a matrix
equation:
w1C1+w2C2+w1C3= 0 (10)
2w1C1+ 2w1C3= 0 (11)
w2C1+w2C2+ 2w1C3= 0.(12)
Answer:
w1w2w1
2w10 2w1
w2w22w1
C1
C2
C3
.= 0 (13)
II. DETERMINANTS
The determinant of the 2 ×2 matrix
A=a11 a12
a21 a22 (14)
