# A to Z of Excel Functions: The MUNIT Function

19 September 2022

*Welcome
back to our regular A to Z of Excel Functions blog. Today we look at the MUNIT function. *

* *

**The
MUNIT function**

In matrix algebra, the **MUNIT** function
returns the unit matrix for the specified dimension. A unit matrix is simply a square (**n **x **n**)
matrix which has the value of one [1] down the leading diagonal (from top left
to bottom right) and zero [0] everywhere else.

Described like this, it doesn’t sound very exciting!

It has the following syntax:

**MUNIT(dimension)**

The **MUNIT **function
has the following argument:

**dimension:**this is required and represents an integer specifying the dimension of the unit matrix that you wish to return, resulting in a square (**n**x**n**) array. This**dimension**must be greater than zero [0].

It uses the following equation:

If **dimension** is a value less than or equal
to zero [0], **MUNIT** returns the *#VALUE!* error value.

If you are using Microsoft 365, then you can simply
enter the formula in the top left-cell of the output range and then press **ENTER **to confirm the formula as a dynamic array formula. Otherwise, the formula must be entered as a
legacy array formula by first selecting the output range, entering the formula
in the top-left-cell of the output range, and then pressing **CTRL + SHIFT +
ENTER** to confirm it. Excel inserts
curly brackets (known as **braces**) at the beginning and end of the formula
for you.

In Excel 365:

In a “legacy” version of Excel:

So what would you use this function for?

Let’s keep this simple, and just consider a 2 x 2
matrix scenario. Matrices are used to
transform data, often used in computer graphics, cartoons, *etc. *Sometimes, you wish to maintain a
character’s direction or bringing it nearer or closer (scaling up or scaling
down).

In linear algebra, the term **eigenvector** is
used to denote the direction that should remain consistent. The corresponding term **eigenvalue **would
denote the scale of the enlargement, *e.g.*

- 1 would mean “no change”
- 2 would mean “double the length”
- -1 would mean point in the opposite direction to the eigenvalue.

Of course, mathematicians (such as myself!) have to complicate this notion in case someone still understands what I am talking about. It may be represented mathematically in the form

**Av = ****λ****v**

where:

**A**is a matrix of dimension**n**x**n****v**is a vector, consisting of one row or one column, with**n**elements. This is the eigenvector**λ**is the scalar multiple (*i.e.*it is simply a number) which represents the eigenvalue.

[For what it’s worth, “eigen” comes from the German word for “typical”, if that helps clarify.]

As an example, let’s consider the 2 x 2 matrix

This has an eigenvector of with an associated eigenvalue of five [5].

How on earth was this calculated? Let’s revert to the original premise:

**Av = ****λ****v**

To ensure I am considering like with like, I will make sure both sides
of the equation are matrices, so I will add the 2 x 2 identity matrix, *i.e. ***MUNIT(2)** here:

**Av = ****λI****v**

where **I **is the unitary 2 x 2 matrix. Thus:

**Av – ****λI****v = 0**

Assuming the eigenvector **v **is non-zero, we may solve for the
eigenvalue **λ **by calculating the determinant of the matrix. This is a special calculation in linear
algebra, denoted by **|A|**, meaning the determinant of the square matrix **A**. For a 2 x 2 matrix, the determinant is
calculated as

To solve for **λ**, we calculate

**|A – ****λI****| = 0**

Therefore,

which reduces to

Calculating this determinant, we get

**(-5 – ****λ)(4 – λ) – (2 x 5) = 0**

Which simplifies to a quadratic equation, *viz.*

**λ ^{2} + λ – 30 = 0**

which is

**(****λ + 6)( λ – 5) = 0**

*i.e. ***λ** equals either -6 or +5. That means there are __two__ distinct
eigenvalues, one of which is five [5].
Knowing this, we may now calculate the associated eigenvector:

Using standard matrix algebra, we get the two equations

**-5x + 2y = 5x**

**5x + 4y = 5y**

Bringing all values over to the left-hand side of the equations, we get

**-10x + 2y = 0**

**5x – y = 0**

Clearly, **y = 5x**, so an eigenvector is any non-zero multiple of
the vector. For example,

*i.e. ***Av = ****λ****v**

If you wanted to calculate this in Excel, you would need **MUNIT**. Further, if you add up the digits on the
leading diagonal (known as the **trace **of the matrix), you will note

-5 + 4 = -1

The two solutions for **λ **are -6 and +5, which also add up to -1.
This is because the sum of the eigenvalues of a matrix equals the trace
of that matrix. This can be a good
check, which **MUNIT **may also be used for:

**trace(A) =SUM( A *
MUNIT(ROWS(A)) )**

So ends today’s mathematics lesson. Class dismissed.

*We’ll continue our A
to Z of Excel Functions soon. Keep
checking back – there’s a new blog post every other business day.*

*A full page of the function articles can be found **here.*