The Spectral theorem establishes conditions under which an operator can be expressed in a simple form ( sum of “simpler” operators). This talk is divided into two parts. In the first talk, we will discuss the spectral theorem in a finite dimensional complex Hilbert space. In a finite dimensional space, the spectral theorem says that a normal linear operator/matrix is unitary equivalent to diagonal operator/matrix (here diagonal matrices are the “simpler” matrices). We will see one important application of the spectral theorem, namely the idea of defining a functional calculus. In the second talk, we will do the same on an infinite dimensional Hilbert space.