Orthogonally Diagonalizable Matrices

1 Orthogonally Diagonalizable MatricesThese notes are about real Matrices Matrices in which all entries are real numbers. Complexnumbers will come up occasionally, but only in very simple ways as tools for learning moreabout real matrix is called if we can write where is a8 8EE THTH Diagonalizable "diagonal matrix. This is possible if and only if there is a basis .. for , ,," #8 8 where the 's are eigenvectors of The corresponding eigenvalues sit along the,3E diagonal of . and the matrix [.. ]. Thus , the change ofHT T T, ,," #8 U coordinates matrix: and T T U UUUBBB B "E acts like a diagonal matrix when we change coordinates: more precisely , themapping (in standard coordinates) is the same as (written inBBBB E H UUU-coordinates).

2 An is a square matrix for which ; , anorthogonal matrixequivalentlyY Y "Xorthogonal matrix is a square matrix with orthonormal An matrix is called 8 8 Eorthogonallyorthogonal Diagonalizable if there is an matrix and a diagonal matrix for which YHE YHY YHY "XThus, an Orthogonally Diagonalizable matrix is a special kind of Diagonalizable matrix: not onlycan we factor , but we can find an matrix that works. In thatE THTY T "orthogonalcase, the columns of form an basis for . We want to know which Matrices areYorthonormal 8orthogonally Diagonalizable .

3 The that appears later in these notes will give usSpectral Theoremthe EE E is called a if symmetric matrixXNotice that a symmetric matrix must be square (?).EwhyExample If is square. is also symmetricE is any matrix (square or not), then E EE EXXbecause E E E E E E X XX XXXThe next result tells us that only a symmetric matrix has a chance to be orthogonallydiagonalizable. This is the easy half of the Spectral If is Orthogonally Diagonalizable , then must be Suppose that is diagonal, orthogonal and.

4 HYE YHY YHY "XThen , so is symmetric. E YHY Y H Y YHY E E XX XXX X XXTo completely understand which Matrices are Orthogonally Diagonalizable , we need to know a bitmore about symmetric Matrices . For instance, a property that symmetric matricescharacterizesis how nicely they interact with the dot An matrix is symmetric for all vectors and8 8EE Eif and only ifB C B CBCin 8 Proof i) Let be in For matrix B C 8 8E 8any (*)E E E EB CB C B C BCXX XX If is symmetric, then and equation (*) becomes.

5 EE EE EXB C B C ii) Suppose for all vectors and in . Let , be theE E B C B CB C++ 8"8columns of . Then for all ,E" 3 4 8 the entry in .E + 4 3 E/ /+ /3 43 4 43 ll the entry in / // +343 4 E + 3 4 E34so is symmetricE The next result about symmetric Matrices uses a few facts about complex complex number has the form where and are real and ForD + ,3 + ,3 " #D + ,3D D + ,3D , the conjugate of is . Clearly, is a real number if and only if, !

6 D D , and this happens if and only if is the set of all complex numbers, and . For the of is (a real number). ClearlyD D lDl + , magnitude # #DD + , lDlD A # ##, and for every , it is easy to check that DA D A ___For a matrix with complex entries, denotes the where each in EE+ E conjugate matrix34has been replaced by . So is a matrix if and only if __+ EE E 34realWe also use the Fundamental Theorem of Algebra (). It tells us thata much deeper result! if we allow complex numbers, then every polynomial factors completely into linear particular, every characteristic polynomial factors completely asG -G 8 8E-- - - -- -"#8 Therefore every matrix has eigenvalues for example, if the factor repeats8 if we count by multiplicities- -3exactly three times, then counts as three is the next important fact about symmetric If is a (real) matrix, then has eigenvalues (E8 8E 8symmetricrealcounted bytheir multiplicities.)

7 For each eigenvalue, we can find a real eigenvector associated with According to the Fundamental Theorem of Algebra, has eigenvalues ,E --"8(possibly with some duplicates listed because we count by multiplicities). Because isEsymmetric, we will show that each must be a real , notice that for the scalar is a any complex vectorrealDD D ; ED D "88X number because ; ; ; E E E E E E ; D D D DD D D D D D D D XXX because is symmetricbecause is real EE Let be an eigenvalue and let be one of its "8D D D Then D D D DD DD D E XXX333.

8 D D D D D D lD l lD l --3 " " # #8 83 "8##But is and, on the right side of the equation, is both D D E lD l lD l X##"8 realrealand nonzero (). Therefore is "8X## E lD l lD l D DSince each is real, is a matrix and det because is an----3333E M E M !realeigenvalue. So the matrix equation has nonzero real solutions In otherreal E M -3B ! words, there are real eigenvectors for eigenvalue -3 We are now ready to prove our main theorem. The set of eigenvalues of a matrix is sometimescalled the of the matrix, and orthogonal diagonalization of a matrix factors in aspectrumEEway that displays all the eigenvalues and their multiplicities.

9 Therefore the theorem is called theSpectral Theorem for real symmetric Spectral Theorem A (real) matrix is Orthogonally Diagonalizable 8 8 EEif and only ifis , we made the easy observation that if is Orthogonally Diagonalizable , then it isEnecessary that be symmetric. The Spectral Theorem says that the symmetry of is alsoEEsufficient: a real symmetric matrix must be Orthogonally Diagonalizable . This is the part of thetheorem that is hard and that seems surprisingbecause it's not easy to see whether a matrix is Diagonalizable at is a proof by induction, and it uses some simple facts about partitioned Matrices and changeof The proof is already half done.

10 We only need to show that a (real) symmetric8 8matrix is Orthogonally is obviously true for every matrix if , then " "E E + E " + " YEY X Assume now that (**) every symmetric matrix is Orthogonally Diagonalizable . 8 " 8 " We will show that (**) it to be true that every symmetric matrix ( the next sizeforces8 8up ) must also be Orthogonally we can do this, we will have finished a proof by induction: because the theorem trueiswhenever is , then it be true whenever is ; then, because itE " "E # # ,?>must alsois true whenever is it be true whenever is ; but then, becauseE # # E $ $must alsoit is true whenever is , it must be true whenever is but then.