Flipping rows into columns, and why the transpose of a product reverses it

0:00The transpose is one move, and it is not a hard one. Whatever sat in row i, column j now sits in row j, column i. That is the whole definition. Every entry keeps its value and changes its address, and the two numbers in that address swap places. Picture the diagonal running from the top left corner down to the bottom right. The transpose reflects the matrix in that diagonal.

0:30An entry sitting on the diagonal has its row equal to its column, so swapping them changes nothing. It stays exactly where it is. An entry off the diagonal crosses to the matching position on the other side. Here is the mistake to get out of the way at the start. Reflecting a matrix in a vertical line is not the transpose. That mirror sends row i, column j to row i, column n plus one minus j. The row never changes. Nothing crosses the diagonal.

1:08For a matrix that happens to be symmetric the two agree, which is exactly why the mistake survives. Try it on almost anything else and it fails. Because the two address numbers swap, the two counts swap with them. An m by n matrix transposes into an n by m matrix. Three rows and two columns become two rows and three columns. Only a square matrix keeps its order, and it keeps it because its two counts were equal to begin with.

1:43So a row matrix, one row and several columns, transposes into a column matrix. Those are different kinds of object, and the transpose is the operation that connects them. That single fact, that the order swaps, is going to do more work in this video than anything else. Take a three by two. Three and five in the first row. The square root of three and one in the second. Zero and minus one fifth in the third.

2:16Its transpose has two rows. The first is the first column read downwards: three, the square root of three, zero. The second is the second column read downwards: five, one, minus one fifth. Three by two became two by three. Every entry survived. Only the addresses moved. Watch minus one fifth in particular. It sat at row three, column two. It now sits at row two, column three. There are two symbols in common use for this. A small prime after the letter, and a raised capital T. They mean the same operation.

2:59This video will use the raised T throughout. One more, quickly. A column holding five, one half and minus one transposes into a row holding five, one half and minus one. Three by one became one by three. Now four properties worth knowing. Three of them fall straight out of the relabelling. The fourth does not, and the fourth is why this topic exists. First: transpose twice and you are back where you started.

3:33Swap the two address numbers, then swap them again. Row i column j goes to row j column i, and then back to row i column j. There is nothing to check here. It follows from the move being a swap. But watch what happens when the move is not a swap. A quarter turn of the whole array also sends rows into columns, and it also flips the order. It looks like a rival.

4:02Do a quarter turn twice, though, and you get a half turn, not the matrix you started with. Across a population of one hundred and forty-five matrices, transposing twice returns every single one. Quarter-turning twice returns seventeen. Second: scaling and transposing pass through each other. Multiply every entry by a number and then transpose, or transpose and then multiply every entry by that number. Same answer either way. The reason is the same reason as before. Scaling does not care about addresses, and transposing does not care about values.

4:48Do the check with the multiplier left as a letter rather than a number. That is the honest version, because a letter cannot accidentally be the one value that makes both sides agree. Over three hundred and twenty-four scaled cases, this holds every time. Third: adding and transposing pass through each other too. Take a two by three holding three, the square root of three and two above four, two and zero.

5:20And a second two by three holding two, minus one and two above one, two and four. Their sum is five, the square root of three minus one, and four; above five, four and four. Transpose that sum and you get the same three by two you get by adding the two transposes separately. Over four thousand and ninety-six pairs, the sum law holds without exception. There is a sharper version of this question, and it is the one that catches people.

5:53Suppose you are handed the transpose of a matrix rather than the matrix itself, and asked to check the difference law. Then you have to undo one flip before you start. Transpose what you were given, work from that, and the difference law comes out as it should. The subtlety is only bookkeeping. But bookkeeping is where the mistakes live. Fourth, and this is the one. Transposing a product does not give the product of the transposes.

6:27It gives the product of the transposes with the factors turned around. Write it out. The transpose of A times B is B transpose times A transpose. B comes first. The version most students write is A transpose times B transpose, in the original order. It is wrong, and it is wrong for a reason worth understanding. Over four thousand and ninety-six pairs that have a product, the reversed law holds every single time.

7:01The straight one holds zero times. Not rarely. Never. So why does the order reverse? Not by convention. By counting. A product needs the columns of the left factor to match the rows of the right one. Write the four counts in a line: m, n, n, p. The inner pair agree, so the product exists, and the outer pair give its order: m by p. Now transpose both factors. The m by n becomes n by m. The n by p becomes p by n.

7:38Put those down in the original order and the line reads n, m, p, n. The inner pair are m and p. They have no reason to agree, so usually there is no product at all. Put them down reversed and the line reads p, n, n, m. The inner pair are n and n. They agree, always, because they agreed in the original product. And the outer pair give p by m, which is exactly the order the transpose of the answer has to have.

8:11So the reversed arrangement is not one of two options that happens to win. It is the only arrangement whose counts fit. Count it. Of the sixteen orders from one by one up to four by four there are two hundred and fifty-six ordered pairs, and sixty-four of them have a product. For all sixty-four, the reversed product of the transposes exists. For only sixteen does the straight one exist at all. And of those sixteen, only four land on the order the answer's transpose has.

8:48Take a two by three times a three by two. The straight product of the transposes does exist. But it comes out three by three, and the transpose of the answer is two by two. Change the right factor to a three by four and the straight version does not even join. The reversed one still does. Here is the cleanest instance of that argument. A three by one column holding minus two, four and five. Times a one by three row holding one, three and minus six.

9:26The inner pair is one and one, so the product exists, and the outer pair makes it three by three. It comes to minus two, minus six and twelve; four, twelve and minus twenty-four; five, fifteen and minus thirty. Transpose that, and you get minus two, four and five; minus six, twelve and fifteen; twelve, minus twenty-four and minus thirty. Now come at it the other way. Transposing turns the column into a row and the row into a column.

10:01Reversed, that is a three by one times a one by three again, and it reproduces the transpose exactly. Straight, it is a one by three times a three by one. That does exist. But it is a single number. One by one, against three by three. There is no reading under which a single number is the transpose of a nine-entry array. One more thing shows up in a second pair of the same shape.

10:31A zero anywhere in the column factor wipes out a whole row of the product, because every entry of that row was built from it. And after transposing, no row of the result is zero. A column is. The wipe-out crossed the diagonal with everything else. Now step back and ask what checking a law on one example actually establishes. Here are six rules. Each one takes a matrix and hands you back another matrix.

11:04Turning rows into columns. Mirroring left to right. Mirroring top to bottom. A quarter turn. Transposing and then squaring every entry. Transposing and then negating every entry. Run all four properties past all six of them. Doing it twice returns you to the start. True for the transpose, for both mirrors and for transpose-then-negate: one hundred and forty-five out of one hundred and forty-five. The quarter turn returns seventeen. Transpose-then-square returns eighty.

11:43Scaling passes through. Three hundred and twenty-four for five of the six. Only one hundred and sixty-four for transpose-then-square. Adding passes through. Four thousand and ninety-six for five of the six, and seven hundred and twenty-nine for the odd one out. And the reversal law. Four thousand and ninety-six for the transpose. Zero for both mirrors. One thousand seven hundred and ninety-two for the quarter turn. Two thousand three hundred and eighty-six for square. Three hundred and forty-three for negate.

12:23Exactly one rule in that table passes all four. The mirrors are the instructive failures. On two by two matrices, mirroring left to right gives the transpose exactly three times out of eighty-one. On the wider ones it does not even get the order right. Not once. That table is doing a job worth naming out loud. The four properties are usually just listed, with a note saying they are given without proof and a suggestion to check them on examples.

13:00Checking them on an example is worth doing. But be clear about what it shows. It shows that the law holds there. Take mirroring left to right, and run it on the worked instances from earlier. Doing it twice returns the matrix: true. Scaling passes through: true. Adding passes through: true. Three properties out of four, passed on the very examples the properties are usually checked against, by a rule that is not the transpose.

13:33The fourth is where it dies. Mirroring left to right never reverses a product correctly. Not once in four thousand and ninety-six chances. So one verification is evidence, and evidence is worth having. The counting argument, three scenes back, is the reason. One last use of the transpose, and it points at everything that comes after it. Ask which matrices satisfy this: the transpose of A, times A, is the identity.

14:08Take the two by two with the cosine of an angle and the sine of an angle in the top row, and minus the sine and the cosine below. Multiply its transpose by itself. Each diagonal position becomes the cosine squared plus the sine squared, which is one. Each off-diagonal position becomes a pair that cancels, which is zero. That is the identity, at every angle. Checked at twenty-one exact angles, all twenty-one work.

14:42A second matrix, the sine and the cosine above minus the cosine and the sine, does the same thing at all twenty-one. Now the trap. Neither of these is the identity matrix. At only two of the twenty-one angles does one of them happen to be the identity. At the other nineteen, neither is. Among the eighty-one two by twos built from minus one, zero and one, eight satisfy the condition. Exactly one of those eight is the identity.

15:17The condition is far weaker than being the identity, and the two are worth not blurring together. To finish, put that condition to work on three unknowns. Take a three by three. First row: zero, twice y, and z. Second row: x, y, and minus z. Third row: x, minus y, and z. Require that its transpose times itself is the identity. Every off-diagonal position vanishes on its own. The terms cancel for any values at all, so those six equations tell you nothing.

15:56The three diagonal positions are what remain. Twice x squared. Six times y squared. Three times z squared. Set each of them equal to one. So x squared is one half, y squared is one sixth, and z squared is one third. Which gives x as plus or minus one over the square root of two, y as plus or minus one over the square root of six, and z as plus or minus one over the square root of three.

16:30Every one of the three carries both signs, and it carries them independently of the other two. So the question has eight answers, not one. And that is the transpose. One relabelling. Row i column j becomes row j column i. Three of its four properties fall straight out of that relabelling. The fourth turns the factors around, and it turns them around because that is the only arrangement whose counts fit.