Can't call getElementsByTagName on loaded svg

I'm loading an SVG using an XMLHttpRequest but I don't seem to be able to parse it. Here's how I'm loading it

    var svgDoc;
    var xhr = new XMLHttpRequest();
    xhr.open("GET", "data/vectors.svg", true);
    xhr.onload = function (e) {
        if (xhr.readyState === 4) {
            if (xhr.status === 200) {
                svgDoc = xhr.responseText;
                console.log(svgDoc.getElementsByTagName("svg"));
            } else {
                console.error(xhr.statusText);
            }
        }
    };
    xhr.onerror = function (e) {
        console.error(xhr.statusText);
    };
    xhr.send(null);

It's throwing Uncaught TypeError: undefined is not a function. When I try console.log(svgDoc.getElementsByTagName) I just get 'undefined'. It's like I can't call any of the XML DOM methods on the SVG - despite the fact that SVG is just XML. Why is this?