An ordered rectangular array, and what its order tells you before anything else

0:00Start with something almost too small to be interesting. One number, written inside a pair of square brackets. Fifteen notebooks. The fifteen is doing the usual work. The brackets are doing something else, and what they are doing is the entire subject. The brackets say: this number is in a position. Right now there is only one position, so the claim looks empty. Watch what happens when there are more.

0:30Now two numbers in brackets. Fifteen and six. Notebooks and pens, in that order, and the order is not decoration. Swap them and you have said something false about somebody's bag. That is the whole idea, and everything else in this video is that idea made precise. A list of numbers carries the numbers. An arrangement carries the numbers and where each one sits. The thing being built is called a matrix. Before we can do anything with one, we have to be able to say what shape it is, and to say it in a fixed order that never varies.

1:13Three people. Each has some notebooks and some pens. The first has fifteen notebooks and six pens. The second has ten and two. The third has thirteen and five. Six numbers. Now lay them out. Here is one way. One horizontal line per person, one vertical line per possession. Three lines down, two across. And here is another. One horizontal line per possession, one vertical line per person. Two lines down, three across.

1:49Look carefully, because this is the moment the subject starts. Those two pictures hold the same six numbers. Not similar numbers. The same six. As a bag of values they are identical, and the checking confirms it. As arrangements they are not the same object at all, and the checking confirms that too. Each is the other laid on its side. In the first, the first person's line reads fifteen and six. In the second, the first person's line also reads fifteen and six, but it runs the other way.

2:28Two arrangements. One body of data. And they are two different matrices. It is worth being precise about what has been gained, because it is easy to feel that nothing has. Suppose I hand you only the six numbers, in no particular order. Fifteen, six, ten, two, thirteen, five. All six are different, so how many ways could they be poured into a frame with three lines down and two across?

3:00Seven hundred and twenty. Every one of them is a perfectly valid arrangement of those six numbers, and exactly one of them is the truth about who owns what. And that is only for one shape. Six entries also fit a frame one across, or six across, or the other way round - four shapes in all. Across all four, two thousand eight hundred and eighty arrangements. So the list narrows you to two thousand eight hundred and eighty possibilities. The arrangement picks one of them.

3:36That gap is the information the arrangement carries and the list throws away. It is not a small amount of information, and it is the reason position is part of the data rather than a way of writing it down. So here is the definition, and it is worth reading slowly because every word in it is load-bearing. A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called its elements, or its entries.

4:12Three words to stop on. Rectangular. Ordered. Array. Rectangular means every horizontal line has the same length as every other. That is not a formatting preference. Watch what breaks without it. Take an arrangement with two lines, one holding two entries and one holding a single entry, and skip the check that would refuse it. Count the entries one at a time and you get three. Read its shape off the page - two lines down, two across - and multiply, and you get four. Those two answers disagree, and an object whose size has two different answers is not something you can do arithmetic with.

4:58So the rectangle is what makes counting and multiplying agree. Every proper matrix is turned away at the door unless it is one. Ordered means the arrangement is fixed, not merely present. And array means the thing itself is the arrangement. A matrix is not a way of writing a number. Nothing here ever collapses one down to a single value. Names, so that we can talk about the parts.

5:30A horizontal line of an array is a row. A vertical line is a column. That is all the vocabulary needed, and it is worth fixing now because everything from here on is stated in it. Here are three arrays to practise on, and they are deliberately varied. The first has three rows and two columns, and holds minus two and five, then zero and root five, then three and six.

6:00The second has three rows and three columns, and its entries include a fraction, a decimal, a surd and a negative number. The third has two rows and three columns, and not one of its entries is a number at all. One plus x. X cubed. Three. Then cosine x, sine x plus two, and tangent x. That third one is the point of the three. An entry does not have to be a number. It can be an expression, or a function. All three of these arrays hold at least one entry that is not a plain number, and none of them is made of plain numbers only.

6:43What they have in common is not what is in them. It is that each one has a definite number of rows and a definite number of columns. Which brings us to the single most useful fact about any matrix, and the one to read before anything else. Its order. The pair of counts: how many rows, then how many columns. Rows first. Always rows first. There is no version of this where it depends on the situation.

7:15So count. The first array has three rows and two columns, and its order is three by two. The second, three rows and three columns: three by three. The third, two rows and three columns: two by three. Notice the first and the third. Three by two and two by three. Those are different orders, and later in this subject they will behave completely differently - there are operations that will simply refuse to run on the pair.

7:49And a refusal like that is decided from the order alone, before a single entry has been looked at. That is why this pair of numbers is read first. The order also tells you how many entries there are, because each of the rows contributes one entry per column. Three by two holds six. Three by three holds nine. Two by three holds six. That was checked on every shape from one by one up to eight by eight - sixty-four shapes, and the counted total agreed with the product in every single one.

8:29Now, how to name one particular entry. Write a, with two small numbers after it. The first says which row. The second says which column. Row first again - the same convention as the order, and for the same reason. Here is an array with three rows and four columns, so twelve entries. The entry at row one, column three is nineteen. At row two, column one, thirty-five. At row three, column three, minus five. At row two, column four, twelve. At row two, column three, five halves.

9:09Now the trap. Row two, column three is five halves. Row three, column two is one. Same two digits in the subscript, different order, and completely different entries. They are not the same position and they are not, in general, the same number. Across every array in this video there are twelve places where you could ask that question, and the two answers coincide at just two of them. And those two are one coincidence seen twice. It is in a headcount of workers at three factories: row one column two is twenty-five, and row two column one is also twenty-five.

9:54But one of those twenty-fives is the women at the first factory, and the other is the men at the second. Equal numbers, entirely different facts. That is what it means to say the two are equal only by accident. Write the general case and the notation does the rest. A one one in the top left corner. A one two beside it. Down the left edge, a two one, a three one, and so on. In the bottom right corner, a m n.

10:27The whole array is then written in shorthand as: the array whose i j-th entry is a i j, where i runs from one to m and j runs from one to n, and both are counting numbers. Out of that array you can take two kinds of slice, and they are the two things the two indices were built to give you. Fix i and let j run, and you have the i-th row: a whole horizontal line.

10:57Fix j and let i run, and you have the j-th column: a whole vertical line. On the array we just used, row two reads thirty-five, minus two, five halves, twelve. Column two reads five, minus two, one. They cross at exactly one place, and that place is row two, column two, which holds minus two. That is what a pair of indices does. Each one picks a line, and the entry is where the two lines meet.

11:30And if you ask for a line that is not there - row four of a three-row array, or column five of a four-column one - the honest answer is nothing at all, not a wrong entry. Now run the counting backwards, which is a better question than it looks. You are told a matrix has eight entries. What shapes could it be? You need every pair of counting numbers whose product is eight. Search them rather than guess.

12:01One by eight. Two by four. Four by two. Eight by one. Four shapes. The common mistake here is to find two. One by eight and four by two, say, and stop - because two fours are two fours. But the pairs are ordered. Four by two and two by four are different shapes, and both count. If you ignored the order, you would say two. Because you do not, the answer is four.

12:30Twenty-four entries permit eight shapes: one by twenty-four, two by twelve, three by eight, four by six, and then the same four reversed. Eighteen permits six. Thirteen permits two. Five permits two. Thirteen and five are prime, and that is exactly why. A prime has nothing to be split into except itself and one, so the only shapes are a single row and a single column. Tested over every number from one to forty: there are twelve primes, all twelve permit exactly two shapes, and no composite number permits only two.

13:13Two special cases. Nine permits three shapes, and one of the three is square - three by three, the only case where the two counts can agree. And one entry permits exactly one shape, one by one, which is where we came in. A single number in brackets really is a matrix. So far every matrix has come from data. They can also come from a rule. Give a rule that takes the two indices and returns a value, and give a frame to fill, and the matrix is determined.

13:49Here is one. Three rows, two columns, and the entry at row i, column j is half the size of i minus three j. Six positions, and the rule is evaluated six times, independently. Row one: one, then five halves. Row two: one half, then two. Row three: zero, then three halves. Notice that the rule is not linear - it has an absolute value in it - and nothing went wrong. The rule does not have to be simple. It has to return one value for each pair of indices, and that is the whole requirement.

14:30Now change it slightly. Keep the halving and the absolute value, but move the three onto the other index: half the size of j minus three i, on a frame with three rows and four columns. It looks like nearly the same rule. Of the six positions the two have in common, the entries agree at two and differ at four. Here is a simpler one: twice i, minus j. Row one reads one, zero, minus one, minus two. Row two: three, two, one, zero. Row three: five, four, three, two.

15:09And to see that the order of the indices in the rule matters as much as anywhere else: swap i and j in that rule, and nine of the twelve entries change. The corner that read minus two now reads seven. One more kind of thing that wants to be a matrix, and it is the one that makes the arrangement feel physical. A point in the plane has two coordinates. So it can be written as a matrix with two rows and one column, or as one with one row and two columns. Both are used.

15:47Now take four points - a quadrilateral. Its corners are at one and zero, three and two, one and three, and minus one and two. Write it wide: two rows and four columns. The top row holds all four first coordinates - one, three, one, minus one. The bottom row holds all four second coordinates - zero, two, three, two. Each column is one corner. Or write it tall: four rows and two columns. Each row is one corner.

16:22Both hold the same eight numbers and both describe the same shape. The third column of the wide one reads one and three. The third row of the tall one reads one and three. Same corner, reached two ways. But they are not the same matrix. Two by four and four by two are different orders, and the check that decides sameness looks at the order first and stops there.

16:50Which is the same lesson as the notebooks, arriving from a different direction: same contents, different arrangement, different object. Two housekeeping rules to close on, and then one small honest oddity. The first is notation. A matrix written in shorthand carries its order with it, so that the shape travels with the object and never has to be recovered by looking. The second is a restriction on what may sit in an entry. From here onwards, entries are real numbers, or functions taking real values.

17:31Which is a narrowing, and worth noticing as one. The definition itself admits more than that. This is a working decision about what we are going to handle, taken deliberately. And here is the oddity. Among the three sample arrays we looked at earlier, there is exactly one entry that is not a real number: it sits in the top left corner of the second one, and it has an imaginary part.

18:00It appeared before the restriction was stated. Nothing later uses it, so nothing breaks. But if you read in order you will meet it, and it is better to say plainly that the restriction governs from the point it is made, rather than pretend the earlier example was never there. So, what to carry away. A matrix is an arrangement, and the arrangement is the information. Its order is a pair of counts, rows before columns, and it is read before anything else because the questions that get refused later are refused on the order alone.

18:42Two indices name an entry, row first. The number of entries is the product of the order, and running that backwards gives you every shape a given number of entries permits - ordered pairs, so both ways round count. Same numbers, different places, different matrix. Everything in this subject rests on that one sentence.