Introduction to Diagonalization

Transcription

1 Introduction to Diagonalization For a square matrix E, a process called diagonalization can sometimes give us more insight into how the transformation B ÈE B works. The insight has a strong geometric flavor, but we will see later that it can also be very useful in applications. The idea is to try to find to a new basis for 8 so that, if we compute in terms of -coordinates, E works like a diagonal matrix H. We would like to find -coordinates and a diagonal matrix H so that is the same as B È E B (a computation in standard coordinates) â same geometric point but named in two different coordinate systems á ÒBÓ È HÒBÓ (a computation is done in -coordinates) The point pictured below at the lower left (named Bor ÒBÓ in the two different coordinate systems) is moved by the transformation to the point pictured at the upper right (named EBin one coordinate system or HÒB Ó in the other coordinate system). Geometrically, the one point is moved to the other point; but we describe that action using two different coordinate systems: as B È E B or as ÒBÓ È HÒBÓ.

2 Changing coordinates: remember that multiplication by the change of basis matrix T converts -coordinates into standard coordinates, and the inverse matrix converts standard coordinates into -coordinates: TÒBÓ ÒBÓ œ B œ T So what we want is to find (if possible) a basis and a diagonal matrix H so that B T H T B œ EB Å l à converts Binto ÒBÓ Ð coordinatesñ Å compute HÒBÓ finally, convert the result HÒ Ó back into standard coordinates B In other words, we want a basis and diagonal matrix H so that E factors as T HT œ E (*) 8 $ The same ideas apply just as well in except that it's harder to draw a picture in (and impossible to draw in 8 for 8 $ÑÞ Definition If there is a basis for 8 and a diagonal matrix H such that the factorization (*) is true, we say that the matrix E is diagonalizable. If we can do this, it's a good thing because (if we compute in coordinates): i) diagonal matrices are so easy to work with, and ii) it's easy to visualize geometrically what a mapping D ÈHD does to DÞ Both i) and ii) can be important in applications. nfortunately, it is not always possible to diagonalize a square matrix E. We'll learn something in these notes about when it is possible, but more details will need to wait until Chapter 5. For now, introducing the basic ideas i) will let us practice with changing bases, and ii) will be good preparation for the lster material in Chapter 5.

3 The first example gives an illustration of why diagonalization is useful. Example This very elementary example is in. Exactly the same ideas apply for 8 8matrices E, but working in with a matrix Emakes the visualization much easier. If H is a diagonal matrix, what does the mapping B È HB do to B geometrically? That is, how does H operate on? To be specific, suppose H œ $. ' When multiply the basis vectors and by H, we just get a rescaling. These vectors are mapped onto scalar multiples of themselves: the scalars involved are the diagonal elements of HÞ H/ œ H œ $ œ $ / ß and B For any B H/ œ H œ œ '/ ', something similar happens: B $ B $B H œ œ B ' B 'B H simply rescales each coordinate of B by a factor of 3 in the first coordinate and and by a factor of 6 in the second coordinate. In this particular example, the diagonal entries of H are both bigger than, so multiplying B by H stretches or dilates each coordinate. This is pictured below (for standard coordinates) where $ ÈH œ ' À

4 % Example For a more general like Eœ ß it's not so clear geometrically & what E does to B. However we can get some insight by doing our calculations in terms of a new basis for. First of all, it's a fact ( just check by multiplying, if you like) that % $ $ $ Eœ œ & ' $ $ Å Å Å œ T H T For now, don't worry about where the matrix T Chapter 5. came from. Such questions come up in T is invertible so, by the Invertible Matrix Theorem, its columns are linearly independent and span. Therefore we can use the columns of T as a new basis for : œö, ß, œö ß Þ ( We looked at this basis as an example in the preceding lecture. ) Then T is the matrix that the textbook calls T œ the change of coordinates matrix from -coordinates to standard coordinates. T ÒBÓ œ B

5 The new basis lets us see more clearly how E operates. ( See also the figure, below) Suppose, for example, that B œ % E œ œ Þ & ( (standard coordinates) We can also calculate EB in a more roundabout way but one which gives us new insight: % $ $ $ E œ œ œ & ' ( $ $ Å Å Å Å Å E T H T B In the second calculation, look at what happens to B, step-by-step, as each of the three matrices, in turn, does its work: ) coordinates $ $ $ $ $ B œ È œ œ %. Then ' % ) $ $ $ $ multiplication by Å Å Å TF converts ÒBÓ the diagonal matrix H ( stretched standard coordinates stretches the new - into -coordinates -coordinates by factors of 3 and 6 (in directions corresponding to, and, ) Finally, œ ) ( Å multiplication by T F converts the ÐstretchedÑ -coordinates back into standard coordinates

6 se the figure below to check each of the preceding steps graphically ß as accurately as you can. $ 1) B œ has -coordinates ÒBÓ œ % 2) Stretching ÒB Ó by factors of $ and ' in the, and, directions gives the point with -coordinates. ) 3) Converting these -coordinates back to standard coordinates ) gives œe ( Þ $ So the effect of E on a point B Ðwhen described in standard coordinates) is the same as the effect of the diagonal matrix H on the same point B (when described in - coordinates). H stretches the -coordinates by a factor of $ in the direction of the positive, -axis and by a factor of ' in the direction of the positive, -axis Þ

7 There is another important observation in this example: look especially at the point B œ, or B œ, Þ Then % ' E, œ œ œ $ œ $ &, $ Again, it's helpful to look at what's going on in more detail, using the same 3-step process ( follow each step using the preceding figure) À % $ $ E, œ œ & ' $ $ $ $ $ ' œ œ œ œ $ œ $ ' $ l l l l l -coordinates stretched converted back into standard of, œ -coordinates coordinates unit vector in the direction, % ' You should analyze the equation E, œ = œ' œ' in the same &, ' way. Thus the matrix E maps each vector, and, to a scalar multiple of itself. Such vectors are called eigenvectors of E; the scalars Ð $ for the vector, and ' for the vector, Ñare called the eigenvalues associated with the eigenvectors, and, Þ The reason we were able to do such a nice analysis of how this matrix E works is that we were able to write down ( using the matrix T that I gave you in the beginning) a very special new basis for a basis whose members are eigenvectors of the matrix E. The general definition is:. Definition A nonzero vector B is called an eigenvector of the 8 8matrix E if EB œ -B for some scalar -. The scalar - is called an eigenvalue of EÐassociated with the eigenvector BÑÞ (It's traditional, in almost all books, to use the Greek letter - ( lambda ) to denote an eigenvalue. Some books call eigenvectors and eigenvalues by the loose translations characteristic vectors and characteristic values. ),

8 In the preceding example there is nothing special about the specific eigenvalues $ and '. They might be any other numbers - and - (where perhaps even - œ - ÑÞ In general, - if E can be factored as E œ T T, where T has columns, ß -,, then, and, are eigenvectors of E with eigenvalues - and - because: We can take œö, ß, as a new basis ( why? ) for, and then calculate - E T -, œ T, œ T - - l l -coordinates of, since, œ,, - - œt œò Ó œ œ,, -,, -, l l back to standard coordinates in -coordinates Similarly, E, œ,. - In this case, the geometric interpretation of how E operates on a point is just as it was earlier: convert the standard coordinates B of the point to -coordinates. Then multiplication by H rescales the first -coordinate by a factor of - and the second by a factor of -. ( Of course if, say, - then the rescaling of the first coordinate is a contraction of the first coordinate of B rather than a stretch ; and if (say) -, the rescaling of the second coordinate also reverses the sign of the second coordinate of B (as well as either stretching or contracting)). Again, E acts like a diagonal matrix H if you calculate in -coordinates. Exactly the same algebraic manipulations and geometric interpretation applies in with an 8 8matrix E when it can be factored as Eœ THT, where His diagonal matrix. This motivates the following definition. To repeat the definition given earlier: Definition Suppose E is an 8 8 matrix. If E can be factored as E œ THT ß where H is diagonal, then we say that E is a diagonalizable matrix. ( It is not always possible to factor E is this way. Based on the geometric notions and ideas used above, can you think of a matrix Ethat isn't diagonalizable? Can you think of a matrix for which there are no eigenvectors?) 8

9 The preceding discussion proves the following theorem (where 8 œ Ñ À - Theorem 1 Suppose E is a diagonalizable matrix, say E œ T T - where TœÒ,, ÓÞ Then i) the columns of T are eigenvectors of the matrix E, and ii) they form a basis for The eigenvalues - and - for these eigenvectors are the scalars found on the diagonal of the corresponding column of H. Moreover, a completely similar argument works for an 8 8matrix Eif EœTHT where His diagonal. Therefore we can say Theorem 1 Suppose E is an 8 8 matrix diagonalizable matrix, say - - â EœT â -$ â T ã ã ã - 8, where T œ Ò,, ÞÞÞ, 8ÓÞ Then i) the columns of T are eigenvectors of the matrix E, and ii) they form a basis for 8. The eigenvalues for these eigenvectors are the scalars found on the diagonal of the corresponding column of H (for example, - for, etc.) It turns out that the converse of Theorem 1 is also true: that is, Theorem 2 If E is an 8 8 matrix and if there is a basis œ Ö, ß, ß ÞÞÞß, 8 for 8 consisting of eigenvectors of E (with corresponding eigenvalues -, -,..., -8Ñ, then E is diagonalizable. Specifically, we can factor EœTHT, where - â - â T œ Ò,, ÞÞÞ, 8Ó and H œ ã ã ã ã - 8

10 First we illustrate the meaning of Theorem 2 with another example, using 8œ again to keep the notation as simple as possible. Then we will give the proof of the theorem in the case 8œ. The general proof for an 8 8matrix is done in exactly same way. ( Not surprisingly, the proof closely parallels the ideas in the example). ( Example Suppose Eœ Þ œö and œö ß is a basis %,, for ( why? ). Since ( & E œ œ œ &, we get that % & is an eigenvector with eigenvalue &. Similarly, ( $ E œ œ œ $, so % ' is an eigenvector with eigenvalue $. Thus, œö, ß, is a basis for consisting of eigenvectors of E. This tells use that E is diagonalizable, because: Let TœÒ,, Óœ. Because the columns of Tare linearly independent, T is invertible. Let H œ &, where the diagonal entries are the $ eigenvalues corresponding to the columns of T ( in the same order ) Theorem 2 asserts that having done this, it now turns out that EœTHT. To illustrate Theorem 2, we now verify that this is true. If we multiply EœTHT on both sides by T(on the right), we see that the equation E œ THT is equivalent to ET œ TH. We check that this is true rather than writing down T À ( & $ ET œ œ and % & ' & & $ TH œ œ $ & '

11 Proof of Theorem 2 ( for the case 8œ ) Suppose E is and that œ Ö, ß, is a basis where, ß, are eigenvectors of E, with corresponding eigenvalues - and -. - Let TœÒ,, Óand Hœ ÞSince, and, are linearly independent, T - is invertible Þ To complete the proof that E is diagonalizable, we show now that EœTHT. As in the preceding example, we just need to verify that ET œ T H, and, to do this, we simply need to remember the definition of matrix multiplication. THœT - œòt T Ó œò Ò,, Ó Ò,, Ó Ó - œò-, -, Ó, and ET œeò,, ÓœÒE, E, Óœ Ò-, -, Ó Å because, of E with eigenvalue -, and, is an eigenvector of E with eigenvalue Þ Therefore ET and T H have the same columns, so ET œ T HÞ - The Big Picture, so far In Chapter 4, we have been discussing vector spaces Z (where Z might not be 8 ). After discussing linear independence and spanning sets in this more general setting, we were led to the idea of a basis œ Ö, ß ÞÞÞß, 8 for Z. From that, the nique Representation Theorem (p. 216) led to the idea of using to define coordinates for each B in Z. The - - coordinate vector ÒBÓ œ if B œ-, -, ÞÞÞ-8, ã 8. The coordinate - 8 vector ÒBÓ is a vector in 8. We saw that the mapping B ÈÒBÓ from Z to 8 is an isomorphism (a one-to-one, onto linear transformation) Þ This mapping preserves all vector space operations and all

12 linear dependency and independency relations. For example, a set of vectors in Z is linearly independent (or dependent) if and only if the corresponding set of coordinate vectors in 8 is linearly independent (dependent). Vector spaces computations in Z are perfectly mirrored by the corresponding coordinate vectors computations in 8 Þ. 8 In the special case when Z œ, we could choose a basis œ Ö, ß ÞÞÞß, 8 different from the standard basis Ö/ ß ÞÞÞß / 8. Each vector B in 8 then gets a new name : its coordinate vector ÒB Órelative to the basis. The matrix T œ Ò, ÞÞÞ, 8Ó is the operator that changes -coordinates into standard coordinates according to the formula T ÒBÓ œ B. Since the columns of T are linearly independent, the change of coordinates matrix T is always invertible and T is the operator that converts standard coordinates into -coordinates: ÒBÓ œ T B. For any invertible 8 8matrix E, the column vectors can be used as a basis for 8 and for coordinates. In that case, T is just the matrix EÞ. If a matrix Ecan be factored into the form THT, where His a diagonal matrix, then E is called diagonalizable because in that case E acts like a diagonal matrix when computations are done relative to the basis that consists of the columns. This led us to the idea of eigenvectors and eigenvalues of the matrix E. We proved that a matrix is diagonalizable if and only if has a basis consisting of eigenvectors of Eand indicated that a completely similar proof works for an 8 8 matrix E operating on 8 Þ The conceptual idea of diagonalization and its relation to a basis of eigenvectors is nicely motivated geometrically and not very hard. You may have noticed, however, that in the preceding examples, a factorization of a given matrix E into THT was given, or a basis of eigenvectors was given so that the matrices T and H could be created. But if you are simply given an 8 8matrix E, then trying i) to find its eigenvalues and eigenvectors ( if it has any at all ), and ii) to determine whether 8 does or does not have a basis consisting of eigenvectors of E are harder questions. Be aware of those questions, but try to ignore your worries about them until we get into Chapter 5. For now just focus on the concept of diagonalizationß what it means ß and how diagonalization is connected to eigenvectors and eigenvalues.