这几天用到了逆矩阵，就在这里总结一下逆矩阵和转置矩阵。

逆矩阵

逆矩阵就是一个矩阵的逆向。比如一个点乘以一个矩阵后得到了一个新的点的位置，如果想通过这个点再获得矩阵转换前的位置，那我们就需要乘以这个矩阵的逆矩阵。
在Three.js里面，我们可以通过new THREE.Matrix4().getInverse(matrix4)方法来获得一个矩阵的逆矩阵。

具有的性质：
可逆矩阵一定是方阵。
如果矩阵是可逆的，那它的逆矩阵具有唯一性。
矩阵A的逆矩阵的逆矩阵，等于它自身。

Three.js获得一个矩阵的逆矩阵：

var m = `new THREE.Matrix4().getInverse(matrix4);

Three.js求逆矩阵源码：

		getInverse: function ( m, throwOnDegenerate ) {

			// based on http://www.euclideanspace.com/maths/algebra/matrix/functions/inverse/fourD/index.htm
			var te = this.elements,
				me = m.elements,

				n11 = me[ 0 ], n21 = me[ 1 ], n31 = me[ 2 ], n41 = me[ 3 ],
				n12 = me[ 4 ], n22 = me[ 5 ], n32 = me[ 6 ], n42 = me[ 7 ],
				n13 = me[ 8 ], n23 = me[ 9 ], n33 = me[ 10 ], n43 = me[ 11 ],
				n14 = me[ 12 ], n24 = me[ 13 ], n34 = me[ 14 ], n44 = me[ 15 ],

				t11 = n23 * n34 * n42 - n24 * n33 * n42 + n24 * n32 * n43 - n22 * n34 * n43 - n23 * n32 * n44 + n22 * n33 * n44,
				t12 = n14 * n33 * n42 - n13 * n34 * n42 - n14 * n32 * n43 + n12 * n34 * n43 + n13 * n32 * n44 - n12 * n33 * n44,
				t13 = n13 * n24 * n42 - n14 * n23 * n42 + n14 * n22 * n43 - n12 * n24 * n43 - n13 * n22 * n44 + n12 * n23 * n44,
				t14 = n14 * n23 * n32 - n13 * n24 * n32 - n14 * n22 * n33 + n12 * n24 * n33 + n13 * n22 * n34 - n12 * n23 * n34;

			var det = n11 * t11 + n21 * t12 + n31 * t13 + n41 * t14;

			if ( det === 0 ) {

				var msg = "THREE.Matrix4: .getInverse() can't invert matrix, determinant is 0";

				if ( throwOnDegenerate === true ) {

					throw new Error( msg );

				} else {

					console.warn( msg );

				}

				return this.identity();

			}

			var detInv = 1 / det;

			te[ 0 ] = t11 * detInv;
			te[ 1 ] = ( n24 * n33 * n41 - n23 * n34 * n41 - n24 * n31 * n43 + n21 * n34 * n43 + n23 * n31 * n44 - n21 * n33 * n44 ) * detInv;
			te[ 2 ] = ( n22 * n34 * n41 - n24 * n32 * n41 + n24 * n31 * n42 - n21 * n34 * n42 - n22 * n31 * n44 + n21 * n32 * n44 ) * detInv;
			te[ 3 ] = ( n23 * n32 * n41 - n22 * n33 * n41 - n23 * n31 * n42 + n21 * n33 * n42 + n22 * n31 * n43 - n21 * n32 * n43 ) * detInv;

			te[ 4 ] = t12 * detInv;
			te[ 5 ] = ( n13 * n34 * n41 - n14 * n33 * n41 + n14 * n31 * n43 - n11 * n34 * n43 - n13 * n31 * n44 + n11 * n33 * n44 ) * detInv;
			te[ 6 ] = ( n14 * n32 * n41 - n12 * n34 * n41 - n14 * n31 * n42 + n11 * n34 * n42 + n12 * n31 * n44 - n11 * n32 * n44 ) * detInv;
			te[ 7 ] = ( n12 * n33 * n41 - n13 * n32 * n41 + n13 * n31 * n42 - n11 * n33 * n42 - n12 * n31 * n43 + n11 * n32 * n43 ) * detInv;

			te[ 8 ] = t13 * detInv;
			te[ 9 ] = ( n14 * n23 * n41 - n13 * n24 * n41 - n14 * n21 * n43 + n11 * n24 * n43 + n13 * n21 * n44 - n11 * n23 * n44 ) * detInv;
			te[ 10 ] = ( n12 * n24 * n41 - n14 * n22 * n41 + n14 * n21 * n42 - n11 * n24 * n42 - n12 * n21 * n44 + n11 * n22 * n44 ) * detInv;
			te[ 11 ] = ( n13 * n22 * n41 - n12 * n23 * n41 - n13 * n21 * n42 + n11 * n23 * n42 + n12 * n21 * n43 - n11 * n22 * n43 ) * detInv;

			te[ 12 ] = t14 * detInv;
			te[ 13 ] = ( n13 * n24 * n31 - n14 * n23 * n31 + n14 * n21 * n33 - n11 * n24 * n33 - n13 * n21 * n34 + n11 * n23 * n34 ) * detInv;
			te[ 14 ] = ( n14 * n22 * n31 - n12 * n24 * n31 - n14 * n21 * n32 + n11 * n24 * n32 + n12 * n21 * n34 - n11 * n22 * n34 ) * detInv;
			te[ 15 ] = ( n12 * n23 * n31 - n13 * n22 * n31 + n13 * n21 * n32 - n11 * n23 * n32 - n12 * n21 * n33 + n11 * n22 * n33 ) * detInv;

			return this;

		}

转置矩阵

说到转置矩阵，这里就要说一下矩阵的排列。矩阵的排列有两种方式：行优先和列优先。而转置矩阵其实就是行优先和列优先之间的切换。
两种方式在数学上没有什么不同，大多数人都习惯上使用行优先。在Three.js中，我们需要通过行优先设置，而在存储中，则使用列优先存储数据到elements。
设置矩阵时使用行优先：

var m = new Matrix4();

m.set( 11, 12, 13, 14,
       21, 22, 23, 24,
       31, 32, 33, 34,
       41, 42, 43, 44 );

元素数组elements将存储为:

m.elements = [ 11, 21, 31, 41,
               12, 22, 32, 42,
               13, 23, 33, 43,
               14, 24, 34, 44 ];

在Three.js中，我们可以同过transpose()方法，将矩阵转置：

matrix.transpose();

Three.js的转置源码：

	transpose: function () {

		var te = this.elements;
		var tmp;

		tmp = te[ 1 ]; te[ 1 ] = te[ 4 ]; te[ 4 ] = tmp;
		tmp = te[ 2 ]; te[ 2 ] = te[ 8 ]; te[ 8 ] = tmp;
		tmp = te[ 6 ]; te[ 6 ] = te[ 9 ]; te[ 9 ] = tmp;

		tmp = te[ 3 ]; te[ 3 ] = te[ 12 ]; te[ 12 ] = tmp;
		tmp = te[ 7 ]; te[ 7 ] = te[ 13 ]; te[ 13 ] = tmp;
		tmp = te[ 11 ]; te[ 11 ] = te[ 14 ]; te[ 14 ] = tmp;

		return this;

	},