The named shapes — row, column, square, diagonal, scalar, identity, zero

0:00Seven names arrive together, and they arrive as a list. Column. Row. Square. Diagonal. Scalar. Identity. Zero. A list is the worst possible way to meet them, because they are not seven separate things. Four of them are one thing, unfolded in four steps, and the other three are cut on a completely different axis. Here is the axis. Every one of these names is the answer to a single question: what does this array have to satisfy before it may be called that? And the answer is always a condition - either on the order, or on the entries, or on both.

0:43Three of the seven ask only about the order. Three ask about the entries as well. One asks only about the entries and says nothing whatever about the shape. By the end of this you will not be remembering seven definitions. You will be reading four conditions stacked on top of each other, and three more standing off to one side. Start with the simplest condition anyone could write down. Exactly one column.

1:16Here is one. Four entries, running down: zero, root three, minus one, one half. Its order, counted rows first, is four by one. Four rows. One column. That is the entire definition, and notice what it does not say. It says nothing about how many rows there are. It says nothing about what the entries are - one of these is zero, one is irrational, one is negative, one is a fraction, and the name does not care about any of that.

1:50In general the shape is m by one, for any m at all. The second count is pinned to one and the first is free. A condition that pins one count and leaves the other free is going to be a recurring shape in this video. Hold on to it. Now pin the other count instead. Exactly one row. Here is one. Four entries, running across: minus one half, root five, two, three. Its order is one by four.

2:24And now the question worth stopping for. Is that the same object as the column we just built, turned on its side? No. And the reason is not subtle, it is the same reason as everything else in this subject. One is four rows by one column. The other is one row by four columns. Those are different orders, so they are different matrices, and nothing in this video will turn one into the other.

2:54There is exactly one place where the two names meet. Search every array with up to three rows and three columns and ask which are both a row and a column: three of them, and every single one has order one by one. A one by one array is a row matrix and a column matrix at the same time. That is not a trick. It is the first sign of something that will happen four more times before this video is finished.

3:26One more warning while both are on the board. Each of these holds four entries. Four is a square number. Neither array is square. Square. The two counts agree. Here is one: three rows, three columns, nine entries. And here is a small thing worth saying out loud. Up to now every shape has needed two numbers to describe it. This one needs one. Once the counts agree, saying three says everything.

3:58Every remaining name in this video is built on top of this one. Diagonal needs square. Scalar needs diagonal, so it needs square. Identity needs scalar, so it needs square too. That is three of the four conditions to come, and all three inherit this one. Which means that from here on, if an array is not square, three of the seven names are already unavailable to it before anyone has looked at a single entry.

4:31Row, column and zero are the three that never ask this question at all. We will come back to why. Before the next name, we need a piece of vocabulary, and it is a piece about positions rather than about arrays. In a square array, the diagonal is the collection of entries whose two indices agree. Position one one. Position two two. Position three three. Top left to bottom right.

5:02On a three by three array holding one, minus three and one; two, four and minus one; three, five and six - the diagonal collects one, four and six. Three entries, at the three positions where the row index and the column index are the same number. Everything else is off the diagonal. And here is the count that will matter more than it looks. At order one there are zero positions off the diagonal. At order two there are two. At order three there are six.

5:37Zero. That is not a rounding of a small number. A one by one array has one position, and its two indices agree, so there is nothing off the diagonal at all. Remember that number. Now the name. A diagonal matrix is a square matrix in which every entry off the diagonal is zero. Read that again, because the commonest mistake anyone makes with this name is to read it backwards. The condition is on the entries OFF the diagonal. Nothing at all is asked of the entries ON it.

6:18So a zero sitting on the diagonal is perfectly legal. A square array of nothing but zeros is a diagonal matrix. Whereas this array, with a zero on the diagonal and a two off it, is not one - and it fails for the two, not for the zero. In index form: the entry at i j is zero whenever i is not equal to j. That is the whole definition and it names the two cases explicitly.

6:50Here are three of them, at orders one, two and three. A single four. Then minus one and two on the diagonal with zeros off it. Then minus one point one, two and three. And the first one is the one to stare at. Why is a single entry, four, a diagonal matrix? Because the condition asks about the off-diagonal positions, and there are zero of them. Every statement about an empty collection is true. It is not diagonal because it looks diagonal. It is diagonal because there is nothing there to violate the rule.

7:32Scalar. A diagonal matrix whose surviving entries all agree. One extra condition on top of the previous name, and only one. Everything off the diagonal is still zero - that is inherited. What is new is that the entries on the diagonal must now all be the same as each other. Notice what is still not being asked. Nothing says what they have to agree on. Three examples, at orders one, two and three: a single three; then minus one repeated; then root three repeated.

8:09One of those repeats a negative number. One repeats an irrational one. Both are scalar matrices, and the definition is completely indifferent. And here is the separator, which is the thing to hold on to. The three by three diagonal example from a moment ago had minus one point one, two and three on its diagonal. Three different values. It is diagonal and it is not scalar, and that single array is the evidence that these two names are not the same name.

8:47Identity. A scalar matrix whose repeated value is one. Again exactly one new condition. Off the diagonal, zero - inherited twice. On the diagonal, all the same - inherited once. And now: the same as what? One. In index form the two cases are written out directly. The entry at i j is one when i equals j, and zero when it does not. On the three by three instance that is three ones on the diagonal and six zeros off it, and both of those counts are exactly what the two cases predict.

9:29The symbol is a capital I, usually with the order written underneath it. Once the surrounding work has fixed what the order must be, the subscript gets dropped, and from that point on the symbol carries the order silently. Now go back to the three scalar examples. Repeated three. Repeated minus one. Repeated root three. How many of those are identity matrices? None of them. Not one. Every scalar matrix except the ones repeating exactly one fails the last condition, and this is the second separator.

10:12Zero. Every entry is zero. That is the entire definition, and read it beside the four we have just built, because something is conspicuously missing. It says nothing about the order. Here are four zero matrices. One by one. Two by two. Two by three. One by two. Two of those four are square and two of them are not, and all four are equally zero matrices. This is the cleanest evidence there is that the seven names are not one family. Search every array up to three by three and ask which of the seven names permit an array that is not square. Three of them do: column, row and zero. The other four do not.

11:01Four names demand squareness. Three do not. And the three that do not are precisely the three that were never part of the chain. One consequence worth stating. Because the order is unconstrained, the symbol for a zero matrix carries no order either. It is written O, and whatever shape the surrounding work needs, that is the shape it has. Now the thing that has been building up since the third minute of this video.

11:34Take the one by one identity - a single entry, one, in brackets. Ask it every one of the seven questions. Is it a column matrix? Exactly one column, yes. A row matrix? Exactly one row, yes. Square? The counts agree, yes. Diagonal? Zero positions off the diagonal, so vacuously yes. Scalar? One entry on the diagonal, and it agrees with itself, yes. Identity? The repeated value is one, yes.

12:08Six of the seven names, carried by one array with one entry in it. The only one it misses is zero, and it misses that for the obvious reason. Swap the one for a zero and it picks that name up too - and drops nothing except identity. The one by one zero also carries six of the seven. This is not a curiosity. It is a demonstration that these are conditions and not pictures. Nobody looking at a single number in brackets would call it diagonal. The condition calls it diagonal, because the condition is about what is off the diagonal, and there is nothing off it.

12:52It also means something for the chain. At order one, three of the four square names collapse into each other. Search every one by one array and ask which of the three links can be told apart: exactly one of them, and it is the last one, scalar against identity. Go up to order two and all three links separate. So the chain is real, but you cannot see it at the smallest size, and the size at which it first becomes visible is two.

13:27So here is the whole thing in one picture, and it is a picture nobody usually draws. Square is the outermost region. Inside it sits diagonal. Inside that, scalar. Inside that, identity. Four nested regions, and each boundary is exactly one extra condition. Let me make that a measurement rather than a drawing. Take every small array whose entries come from zero, one and minus one - eleven different orders, twenty-one thousand four hundred and fifty-nine arrays in all.

14:05Run all seven conditions over every one of them. A hundred and twenty are columns. A hundred and twenty are rows. Nineteen thousand seven hundred and sixty-seven are square. Thirty-nine are diagonal. Nine are scalar. Three are identities. Eleven are zero matrices. Now the containment. How many are diagonal and not square? Zero. How many are scalar and not diagonal? Zero. How many are identities and not scalar? Zero. Three zeros, and those three zeros are the chain.

14:46But a chain of equalities would give three zeros too, so run it the other way. How many are square and not diagonal? Nineteen thousand seven hundred and twenty-eight. Diagonal and not scalar? Thirty. Scalar and not an identity? Six. Zero one way and a large number the other way, at every link. That is what it means for the containment to be proper rather than a rewording. And the same three counts come out zero over a completely different collection - six hundred and eighty arrays, over a different alphabet of entries, up to order two. The chain is not an artefact of the numbers we happened to choose.

15:32One last thing about the picture. Row, column and zero are not rings in it at all. They cut across it. The one by one zero matrix sits at the dead centre of every ring and also carries the name that has nothing to do with any of them. Two questions, and they test the definitions rather than compute with them. First. What condition makes a matrix square? Sweep every order from one by one up to four by four - sixteen of them - and try the candidates.

16:08The two counts agree: that picks out four orders, and they are exactly the four square ones. Now a tempting wrong answer: the number of entries is a square number. That picks out six orders, and two of them are not square at all - one by four and four by one. Four entries, not a square array. Which is precisely the trap we walked past at the start, when the row matrix and the column matrix each held four entries.

16:39Second question. How many three by three matrices are there with every entry either zero or one? Nine positions - counted, not multiplied. Two choices at each. The choices are independent, so generate them all and count them: five hundred and twelve. The tempting wrong answers are twenty-seven and eighty-one, and it is worth seeing exactly what each one confuses. Twenty-seven is three cubed and eighty-one is three to the fourth. Both use three - the order - as the base, when the base should be the number of choices per position, which is two. And both use a small exponent when the exponent should be the number of positions, which is nine. Two to the ninth is five hundred and twelve.

17:30And while those five hundred and twelve are in front of us, run the seven conditions over them. All five hundred and twelve are square. Eight are diagonal. Two are scalar. One is an identity. One is the zero matrix. The chain again, in miniature, on a collection built for a completely different reason. What to carry away. Seven names, and one question behind all of them: what condition does an array have to satisfy?

18:04Four of them nest, each adding exactly one condition to the one before. Square, then everything off the diagonal is zero, then the survivors agree, then what they agree on is one. Every one of those containments is proper, and each has a two by two or a one by one array standing between it and the next. Three of them stand outside: row and column pin one count and free the other, and zero asks nothing about the shape at all.

18:37The smallest array in the subject carries six of the seven names, and it does so because these are conditions and not pictures. And the reason any of this is worth the trouble is what comes next. Whether two matrices may be added is decided on the order. Whether they may be multiplied is decided on the order. Which matrix leaves another unchanged under multiplication is the identity, and the reason it does that is the two-case condition we wrote down for it.

19:13Not seven things to memorise. Four conditions stacked, and three standing aside.