z-space replaces multiplication with convolution and frequency eigenvectors
There may be a confusion here about how to cascade filters versus how to actual perform filtering. 

There many different ways to implement linear time invariant filtering. If the filter can be represented as an FIR filter than two of the choices are time-domain convolution or frequency domain filtering such as overlap add. The later does indeed just require complex multiplication in the frequency domain (plus the actuall FFTs as well). For large filter length frequency domain processing is a lot more efficient but induces significant latency. 

There are many, many other ways to implement filters as well. 

Not sure what you mean by "diagonal matrix multiplication". For a system with a single input and a single output, this is just a simple vector multiply and there is no need for a matrix. While you can express time domain convolution as a matrix multiplication, it's not particularly efficient to actually implement it this way, so the matrix representation is mainly used for mathematical analysis.

EDIT: Lets use a specific example

Let's say you have a real signal of 1024 points and a  real FIR filter of 1024 taps. 
Doing a direct convolution takes (roughly) 1024*1024 multiply-add operations for a total ca. 2.1 million scalar operations.

Doing it in the frequency domain requires a forward FFT, complex multiplication in the frequency domain and an inverse FFT. A forward FFT of length 1024 takes about 5*1024*log2(1024)= 51200 scalar operations. A complex multiply takes about 6 scalar operations. So the total would be 2*51200+6*1024 or roughly 110k operations. In this particular example the frequency domain multiplication is about 20 times more efficient than the time domain convolution. 

DISCLAIMER: This example glosses over a lot of details of the frequency domain method, in particular zero-padding, doing efficiently complex FFTs of real valued signals and overlap handling. However, in terms of computational complexity these mostly cancel each other and don't make a big difference in total operation count.