Kenneth Kuttler

19.8. DIAGONALIZATION OF SYMMETRIC MATRICES 423

Theorem 19.8.5 Let v1 be a unit vector (|v1|= 1) in Rp, p > 1. Then there existvectors {

v2, · · · ,vp}

such that{v1,v2, · · · ,vp

}is an orthonormal set of vectors.

Proof: Use Theorem 18.5.10 to extend {v1} to a basis for Rn and then use Theorem19.8.4.

Thus, as observed above, the matrix(v1 · · · vp

)is a orthogonal matrix. With this

preparation, here is a major result. It is actually a specialization of a much more interestingtheorem. See any of my linear algebra books under the topic of Schur’s theorem.

Theorem 19.8.6 Let A be a real symmetric matrix. Then there is an orthogonaltransformation U such that

UT AU = D

where D is a diagonal matrix having the real eigenvalues of A down the diagonal. Also,the columns of U are an orthonormal set of eigenvectors.

Proof: This is obviously true if A is a 1× 1 matrix. Indeed, you let U = 1 and it allworks because in this case A is already a diagonal matrix. Suppose then that the theoremis true for any k < p and let A be a real p× p symmetric matrix. Then by the fundamentaltheorem of algebra, there exists a solution λ to the characteristic equation

det(A−λ I) = 0.

Then since A−λ I has no inverse, it follows that the columns are dependent and so thereexists a nonzero vector u such that (A−λ I)u= 0 and from Proposition 19.8.1, λ is real.Dividing this vector by its magnitude, we can assume that |u| = 1. By Theorem 19.8.5,there are vectors v2, · · · ,vp such that

{u,v2, · · · ,vp

}is an orthonormal set of vectors. As

observed above, ifU =

(u v2 · · · vp

)it follows that U is an orthogonal matrix. Now consider UT AU. From the way we multiplymatrices, this is

vT2...vT

A(u v2 · · · vp

uT

vT2...vT

( Au Av2 · · · Avp)

uT

vT2...vT

( Au Av2 · · · Avp)=

uT

vT2...vT

( λu Av2 · · · Avp)

Now recall the way we multiply matrices in which the i jth entry is the product of the ith

row on the left with the jth column on the right. Thus, since these columns of U areorthonormal, the above product reduces to something of the form(

λ aT

0 A1

)

19.8. DIAGONALIZATION OF SYMMETRIC MATRICES 423Theorem 19.8.5 Let v; be a unit vector (\v;|= 1) in R?, p> 1. Then there existvectors{2.7 sep}such that {v1 ,U2,°°° ,Up} is an orthonormal set of vectors.Proof: Use Theorem 18.5.10 to extend {v,} to a basis for R” and then use Theorem19.8.4. IThus, as observed above, the matrix ( Vi t+ Up ) is a orthogonal matrix. With thispreparation, here is a major result. It is actually a specialization of a much more interestingtheorem. See any of my linear algebra books under the topic of Schur’s theorem.Theorem 19.8.6 Let A be a real symmetric matrix. Then there is an orthogonaltransformation U such thatU™AU =Dwhere D is a diagonal matrix having the real eigenvalues of A down the diagonal. Also,the columns of U are an orthonormal set of eigenvectors.Proof: This is obviously true if A is a 1 x 1 matrix. Indeed, you let U = 1 and it allworks because in this case A is already a diagonal matrix. Suppose then that the theoremis true for any k < p and let A be a real p x p symmetric matrix. Then by the fundamentaltheorem of algebra, there exists a solution A to the characteristic equationdet (A—AI) =0.Then since A — AJ has no inverse, it follows that the columns are dependent and so thereexists a nonzero vector wu such that (A — AJ) w = O and from Proposition 19.8.1, A is real.Dividing this vector by its magnitude, we can assume that |w| = 1. By Theorem 19.8.5,there are vectors v2,--- ,U, such that {u,V>, a ) v} is an orthonormal set of vectors. Asobserved above, ifU = ( WU v2 +: Vp )it follows that U is an orthogonal matrix. Now consider U7 AU. From the way we multiplymatrices, this isul uPv3 v3: A( u V2 +8 vp )= , ( Au Av2 ++ Av, )uf vlul ulvy vy= . ( Au Avr --- Avp \= : ( Au Avr +--+ Avy )ul ulNow recall the way we multiply matrices in which the ij” entry is the product of the i”row on the left with the jin column on the right. Thus, since these columns of U areorthonormal, the above product reduces to something of the formA a0 A;