2×2矩阵的行列线性变换

An orthogonal matrix is a matrix such that all of its rows and columns are mutually perpendicular unit vectors.

Suppose that we have an orthogonal $2\times 2$ matrix: $$A=\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right]$$ Then the following conditions hold: $$a^2+b^2=1$$ $$c^2+d^2=1$$ $$ac+bd=0$$ $$a^2+c^2=1$$ $$b^2+d^2=1$$ $$ab+cd=0$$ The first three rules come from looking at the row vectors, while the next three come from looking at the column vectors. However, it can be shown that one of these sets of conditions will actually imply the other. Thus, we only need three conditions for a $2\times 2$ orthogonal matrix.

Let us multiply $A$ by its transpose. This will provide us with an alternative definition of an orthogonal matrix: $$AA^T=\left[ \begin{array}{ccc} a & b \\ c & d \end{array} \right]\left[ \begin{array}{ccc} a & c \\ b & d \end{array} \right]$$ $$AA^T=\left[ \begin{array}{ccc} a^2+b^2 & ac+bd \\ ac+bd & c^2+d^2 \end{array} \right]$$ $$AA^T=\left[ \begin{array}{ccc} 1 & 0 \\ 0 & 1 \end{array} \right]$$ $$AA^T=I$$ You can confirm that $A^TA$ yields the same result. Thus, an orthogonal matrix is a matrix whose transpose is its inverse. Although I will not prove it here, this also holds for larger matrices, not just $2\times 2$ matrices.

Before moving on, read the following closely related Expii article concerning the classification of $2\times 2$ orthogonal matrices: Classification Of 2 x 2 Orthogonal Matrices

In the article, I showed that every $2\times 2$ orthogonal matrix has one of the following two forms: $$A=\left[ \begin{array}{ccc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right]$$ $$A=\left[ \begin{array}{ccc} \cos\theta & \sin\theta \\ \sin\theta & -\cos\theta \end{array} \right]$$ Calculating the determinant of $A$, we see that the determinant of an orthogonal matrix is always $1$ or $-1$! This is yet another result that holds for all orthogonal matrices, not just $2\times 2$.

Let us interpret this in the context of linear transformations. Consider the linear transformation defined by $A$.

Since $|\det(A)|=1$, the areas of figures in the plane are unchanged when mapped by the transformation. Thus, orthogonal matrices preserve area.

Notice that in the first matrix, $\det(A)=1$. Since $\det(A)\gt 0$, the orientation of some curve in the plane is preserved when mapped by the transformation $A$. On the other hand, in the second matrix $\det(A)=-1\lt 0$. Thus, the orientation is reversed. From the article mentioned above, we know that the first matrix represents a rotation, and the second is a reflection. This agrees with what we already know about rotations, reflections, and their effect on orientation.