11.4. SCHUR’S THEOREM 195

Proof: From Schur’s theorem above, there exists U unitary (real and orthogonal if A isreal) such that

U∗AU = T

where T is an upper triangular matrix. Then from the rules for the transpose,

T ∗ = (U∗AU)∗ =U∗A∗U =U∗AU = T.

Thus T = T ∗ and T is upper triangular. This can only happen if T is really a diagonalmatrix having real entries on the main diagonal. (If i ̸= j, one of Ti j or Tji equals zero. ButTi j = Tji and so they are both zero. Also Tii = Tii.)

Finally, letU =

(u1 u2 · · · un

)where the ui denote the columns of U and

D =

λ 1 0

. . .

0 λ n

The equation, U∗AU = D implies

AU =(

Au1 Au2 · · · Aun

)= UD =

(λ 1u1 λ 2u2 · · · λ nun

)where the entries denote the columns of AU and UD respectively. Therefore, Aui = λ iuiand since the matrix is unitary, the i jth entry of U∗U equals δ i j and so

δ i j = uTi u j = uT

i u j = (ui,u j)

This proves the corollary because it shows the vectors {ui} form an orthonormal basis. Incase A is real and symmetric, simply ignore all complex conjugations in the above argu-ment.

Finally suppose that U∗AU = D where D is real and diagonal. Thus D∗ = D. Then

A =UDU∗

Thus A∗ =UD∗U∗ =UDU∗ = A. This last uses the fact that (AB)∗ = B∗A∗. ■

Example 11.4.8 Here is a symmetric matrix which has eigenvalues 6,−12,18

A =

 1 −4 13−4 10 −413 −4 1

Find a matrix U such that UT AU is a diagonal matrix.