Why cold changes the difficulty and not the strength
Every other tea brewed cold is a slow extraction: you are pulling soluble compounds out of leaf, and cold does it more slowly and more selectively. Matcha is not brewed at all. The powder is suspended whole in the water and you drink the leaf, so nothing needs to be extracted and the strength does not depend on temperature. What cold does change is wetting. Dry matcha particles resist water, and cold water has a harder time breaking that resistance than hot water does, so cold-whisked matcha clumps more stubbornly and separates faster once it stands. That is a mechanical problem with mechanical answers.
What works
SIFT, always, and more carefully than for hot. Sifting is not optional cold. MAKE A PASTE FIRST. A teaspoon or two of water worked into the powder with the whisk or a spoon until it is a smooth paste, then the rest of the cold water added. This one step solves most cold-matcha complaints. OR SHAKE IT. A sealed jar or cocktail shaker with the powder, the water and ice, shaken hard for ten or fifteen seconds, disperses matcha better than a chasen does in cold water. Japanese convenience stores have sold shaken cold matcha in bottles for years and the technique is the same. DO NOT USE ICE AS THE WATER. Whisk or shake with cold water first and pour over ice, rather than whisking into a glass full of ice — the powder sticks to the cubes. EXPECT IT TO SETTLE. Cold matcha separates within a few minutes because nothing is dissolved. Stir or swirl it again; it is not broken.
It tastes different, and mostly in one direction
Cold suppresses aroma sharply, and matcha's aroma is a substantial part of what an expensive one is for. Cold also reduces perceived bitterness and makes sweetness read more clearly, which is why cold matcha is often more approachable and why a merely decent matcha shows better cold than hot. The practical consequence: cold is not the preparation to open a first-flush single-cultivar tin for. It is an excellent use for a good mid-grade one.
Cold-brewed matcha, which is a third thing
Leaving matcha to stand in cold water for hours in the refrigerator is sometimes described as cold brewing. It is not doing what cold brewing does for leaf tea — there is nothing to extract slowly — and the main effect of the time is that the powder settles out. If the result seems smoother, that is largely because you have decanted a weaker suspension off the top. The genuine cold-water technique for Japanese green tea is kōridashi and ice steeping, and those are for leaf. They are worth knowing and they are a different practice.