Abstract Linear Algebra 48 | Proof of Spectral Theorem

Normal matrices over complex numbers are exactly the matrices that can be diagonalized by a unitary change of basis, and the proof hinges on showing that any “Schur form” of a normal matrix collapses from upper triangular to diagonal.

First comes the easy direction: if a matrix A is unitarily diagonalizable, so that U*AU = D for some diagonal D and unitary U, then A must be normal. The reason is algebraic and local to the diagonal. Since D is diagonal, it commutes with its adjoint: D*D = DD*. Writing A in terms of U and D, one computes A*A and AA* and finds both reduce to U(D*D)U* and U(DD*)U*, which match because D*D = DD*. This establishes that unitary diagonalizability implies normality.

The harder direction starts with a normal matrix A and uses Schur decomposition. Because A is square, there exists a unitary U such that U*AU = R, where R is upper triangular. The key move is to show that normality transfers to R: if A*A = AA*, then the same equality holds for R, since the unitary factors cancel in the corresponding products. So the problem becomes: what do normal upper triangular matrices look like?

A 2×2 calculation gives the pattern. Let R be upper triangular with entries a, b, c, so R = [[a, b],[0, c]]. Computing the (1,1) entry of RR* and R*R yields expressions involving |a|^2 and |b|^2. Equality of the two products forces |b|^2 = 0, hence b = 0. That eliminates the only possible off-diagonal entry in a 2×2 upper triangular matrix, proving that a normal 2×2 upper triangular matrix must be diagonal.

The general case follows by induction on the size n. Assume every normal n×n upper triangular matrix is diagonal. For an (n+1)×(n+1) normal upper triangular matrix R, the first column below the (1,1) entry is already zero by triangularity, so R can be written in block form: R = [[a, W*],[0, R̃]], where W is an n-dimensional column vector and R̃ is an n×n upper triangular block. Normality forces the (1,1) entries of RR* and R*R to match. Those entries become |a|^2 + ||W||^2 on one side and |a|^2 on the other, so ||W||^2 must be 0 and therefore W = 0. With W gone, R̃ remains normal and upper triangular, so the induction hypothesis makes R̃ diagonal. Putting the blocks together shows R itself is diagonal.

Since R is diagonal in the Schur form, U*AU is diagonal, meaning A is unitarily diagonalizable. The spectral theorem is thus proven for normal matrices: normality is equivalent to unitary diagonalizability. A major consequence highlighted at the end is that self-adjoint matrices are a special case of normal matrices, so they too are unitarily diagonalizable—setting up the most important diagonalization used later in the course.