Transpose, Symmetric and Skew-symmetric Matrices

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle first. Fill a three by three matrix with zeros, ones and twos. Nine places, three choices each: three to the power nine, nearly twenty thousand matrices, right? Now one condition: the matrix must look exactly the same in a mirror along its main diagonal. How many now? Today you will count them in two lines, using transposes and symmetric matrices. Questions on these come up often in JEE, so let's begin.

0:33Transpose: rows become columns

So, what is the transpose? Really simple: rows become columns. The small capital T at the top right is a superscript, read as A transpose; some books write A prime, with a small dash. In the example, the big square brackets hold the entries. The first row, one, two, becomes the first column, and zero, three becomes the second column. With me? In symbols: the i j entry of A transpose equals a j i. The small i j below is a subscript, row i, column j, and the round brackets say, take A transpose first. The two indices simply swap. Clear? Now tell me. Suppose A is two by three. The cross is read as by: two rows, three columns. What is the order of A transpose? Three by two! Yes, exactly. So if A is m by n, then A transpose is n by m; the double arrow means implies. Only a square matrix keeps its order. Got it?

1:36Rules of transpose: watch the order

Now the rules. Transpose twice and you are back home: A transpose, transpose, is A again, okay? For a scalar k, just a number, the transpose of k A is k times A transpose. And for the same order, the transpose of a sum is the sum of the transposes. Right? Now the important one. What is the transpose of the product A B? Just A transpose times B transpose, right? No, no, careful! That is the trap. The order reverses: the transpose of A B is B transpose times A transpose, whenever A B is defined. Like socks and shoes: socks go on first, but at night, shoes come off first. That is the reversal law. Clear? And it extends. The transpose of ABC is C transpose, B transpose, A transpose, fully reversed. And for a square matrix, the transpose of A to the power n is A transpose, to the power n. Keep it in mind!

2:39Symmetric and skew-symmetric matrices

Now, imagine a mirror along the main diagonal of a square matrix. If the matrix looks the same in it, it is symmetric: A transpose equals A, so a i j equals a j i, okay? In the example, h sits in row one, column two, and again at its mirror position. Same for g and f. Right? Next, skew-symmetric: A transpose equals minus A, so a i j equals minus a j i. Each entry comes back from the mirror with its sign flipped. Both need a square matrix, since A transpose must have the same order as A, okay? Now think. The diagonal entries sit right on the mirror. In a skew-symmetric matrix, what can they be? Only zero! Yes, exactly. Put i equal to j: then a i i equals minus a i i, and only zero equals its own negative. So the whole diagonal is zero. See the example: zeros on the diagonal, p, q, r above, and their negatives below. Got it?

3:45Every square matrix splits into two

Now a beautiful result. Every square matrix A is P plus Q, where P is one half of A plus A transpose, and Q is one half of A minus A transpose. The small fraction is one over two. Add them: the A transpose parts cancel, and we get A back. Simple! Now, P should be symmetric, and Q skew-symmetric. How would you check that P is symmetric? Take its transpose, yes! By our rules, P transpose is one half of A transpose plus A, which is P itself. Symmetric! The same way, Q transpose comes out as minus Q, so Q is skew-symmetric. With me? And the split is unique. The capital O is the zero matrix. If a matrix equals both its transpose and minus its transpose, it equals its own negative, so it must be the zero matrix. That is what makes the split unique. Clear?

4:42Facts used again and again

Now, facts you will use again and again. Take A times A transpose. By the reversal law, its transpose is A transpose, transposed, times A transpose, and that is A A transpose again. So it is symmetric, and so is A transpose times A, okay? Next, if A is skew-symmetric, the transpose of A to the power n is minus A, all to the power n, that is, minus one to the power n, times A to the power n. So even powers are symmetric, odd powers skew-symmetric. Nice, right? Quick question. If A and B are both symmetric, of the same order, is A B always symmetric? Oh no, careful! Not always. The transpose of A B is B transpose A transpose, which is B A, and B A need not equal A B, right? So A B is symmetric if and only if A B equals B A; the double-headed arrow means if and only if.

5:43Example 1: split A into symmetric plus skew-symmetric

Let's start with an easy one. Matrix A has first row three, five, and second row one, minus one. Write it as a symmetric matrix plus a skew-symmetric matrix. Pause the video, and try it first. Done? Okay, let's check. First, A transpose: rows become columns, so we get three, one, five, minus one. Right? Now for P, one half of A plus A transpose. Adding entry by entry, A plus A transpose is six, six, six, minus two. Oh no, careful! Some students stop here and forget the half. Half of it is three, three, three, minus one, with equal off-diagonal entries. Symmetric, as promised! Next comes Q, the skew-symmetric part. Before computing it, tell me: what must its diagonal entries be? Zero, yes! Now, A minus A transpose is zero, four, minus four, zero, and half of it is zero, two, minus two, zero. Diagonal zero. So the answer: three, three, three, minus one, plus zero, two, minus two, zero. Add them, and our matrix comes back. Correct!

7:03Example 2: count 3x3 symmetric matrices, entries 0, 1, 2

Now, our puzzle, with one more part. How many three by three symmetric matrices have entries from zero, one and two? The curly brackets hold the allowed numbers. And how many have trace three, meaning diagonal entries adding up to three? Pause the video and try it. Done? Okay, let's check. Nine places, three choices each, so three to the power nine, right? No, no, wait! Each entry below the diagonal is a mirror copy of one above it, so it is not free. The free entries are the three on the diagonal, plus the three above it: six. Okay? Each has three choices, so three to the power six: seven hundred and twenty nine. Not twenty thousand at all. Puzzle solved! Now trace three. The round brackets list the diagonal from top to bottom. It can be one, one, one, or zero, one, two in any of its six orders. So seven diagonals. Careful: don't count zero, one, two just once! The three entries above the diagonal stay free: three to the power three, twenty seven ways. The cross here means times, so the total is seven times twenty seven: one hundred and eighty nine. Each part took just two lines!

8:26Quick revision: transpose rules, symmetric, skew, the split

So, quick revision. Transpose: rows become columns, and the transpose of a product reverses the order. Symmetric: A transpose equals A, okay? Skew-symmetric: A transpose equals minus A, with zeros on the diagonal. And every square matrix splits, in exactly one way, into a symmetric part, one half of A plus A transpose, and a skew-symmetric part, one half of A minus A transpose. Now go and try the practice problems on this page. All the best!