# Instrumentation & Measurement Magazine 26-1 - 9

```The first two columns of M−1
contain the essence of the
definition of the second, s (adopted in 1967) and the meter,
m (adopted in 1983), as shown in the box, where it is concluded
that the definitions of the second and the meter did
not change in 2019. The kg column defines the kilogram to
be
1 kg . The proportionality relation becomes an
hv / c 2
Cs
equality by dividing each quantity on the right-hand side by
its exact numerical value [5], which is given in Table 1. The numerical
factor (truncated here for convenience) is:
1 kg 1.476 10 hv / c2
   Cs
40
(7)
This is the definition of the kilogram found in section 2.3.1 of
[2]. The definition is realized by methods where the mass of an
object is determined by traceability to the three defining constants
shown in (7) [2], [10]. Note that the matrix technique
defines all base units simultaneously [5].
Matrix M−1
Take the top-left corner of Fig. 1 as a simple example of
using matrices to define the SI second and meter.
s^ 1 1
M  
m^ 0 1
vcCs

s



First calculate the inverse of M using a function such
as MINVERSE in Excel:
m
M1  

 ^1 1
c ^ 01
v
Cs 



can also be used to find the definition of any
SI unit that can be written in terms of the subset of four chosen
base units (1) [8]. The same results are obtained if, for
example, the joule (J) had been chosen as a base unit instead
of the kilogram. The definitions of the kilogram and all other
units are exactly the same even when the joule functions
as a " base " unit from which the kilogram definition is " derived. "
However, the inverse of the matrix M is still required
to define all SI units. Only square matrices can be inverted,
but not all square matrices can be inverted. Since it is the inverted
matrix that is used to define all SI units in terms of the
selected base units, this places an automatic constraint on
any selection of base units. Aside from that, any set of base
units can be chosen. When the joule plays the role of a base
unit in Fig. 1, the column of labels substitutes J^ for kg^.
Therefore, the third column of the yellow matrix contains the
exponents in the product s1
m0J1
A0
because the Planck constant
has the unit J s, as shown in Table 1. After the matrix is
inverted, the joule remains defined by
J hv Cs
 and (7) remains
the definition of the kilogram. Although the choice of
base units is not unique, the definitions of all SI units defined
by Table 1 are indeed unique.
To use matrix M−1
of Fig. 2 to define any arbitrary unit
that can be written in terms of the chosen base units (normally
the historical base units) [8], one begins with the
desired product of powers of base units as shown in (1). The
exponents are written in a column. An example will make
the process clear. In Table 4 of [2], the ohm (symbol: Ω) appears
as a special name for the following product of four
base units:
Ω kg ms A

23 2
(8)
which is another way of expressing that ampere squared, multiplied
by the ohm, equals the watt.
February 2023
The yellow and gray matrices are identical in this
example, but the meaning of the columns and rows is
not the same. In general, an inverse matrix is different
from the starting matrix. The difference between Fig. 4
and Fig. 5 provides a striking example. By matrix multiplication
[9], which is implemented in Excel by the
MMULT function, it can be verified that the product of
M and its inverse M−1
MM    

1
 11 11 1
 01 01 0
  
 

is the identity matrix, I:
 
   
 
  1  I
 

The inverse matrix is used for the next steps;
next steps would be impossible if M could not be inverted.
Column 1 of M−1
1 s  , from which the SI second is defined to
be   = 9 192 631 770/ΔvCs

Cs
1 s  /
Cs
SI
}SI
Cs
. This is the
definition of the second found in [2] and is numerically
identical to the 1967 definition (one second equals
{ΔvCs
ticks of a cesium atomic clock).
Column 2 of M−1
1  , from which the meter is defined to be
1 m = c/{c}SI·{ΔvCs
m Cs
 c

1
}SI
/ΔvCs
≈ (30.663 319)·c/ΔvCs
. This is
the definition of the meter found in [2] and is numerically
identical to the 1983 definition (one meter equals
the distance light travels in a vacuum during 1/{c}SI
seconds).
The column matrix, V, of the various exponents is shown in
gray in Fig. 3, where the labels to the left of V are a reminder of
the source of the exponents appearing in (8).
Simple matrix multiplication is then required to obtain the
definition of one ohm. This can either be done by hand [9] or
using packaged software-for example, the function MMULT
in Excel. The matrix multiplication M−1
·V results in powers of
the four defining constants in the order they appeared in the
columns of M−1
. The result is:  Cs
again h and e2
{h}SI and e , respectively, to establish equality to the ohm. In
this case, the natural quantity h/e2
2
SI
von Klitzing constant, symbol: RK
2
IEEE Instrumentation & Measurement Magazine
has a name and symbol: the
[6], [7]. Its SI value is now
exact but is approximately he/ 25813 Ω. The definition of

9
v c he he

00 1 2
 
/ Ω, where
2
must be divided by their exact numerical values
contains the information
1
contains the information
```

# Instrumentation & Measurement Magazine 26-1

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