Matrix Tangent Function: Definition, Examples, and Practice
The tangent of a matrix generalizes the scalar trigonometric tangent function to square matrices. It is defined using the matrix sine and cosine functions:
$$ \tan(A) = \sin(A)\cos(A)^{-1} $$In practical computation, the tangent of a matrix is obtained using its power series expansion:
$$ \tan(A) = A + \frac{A^3}{3} + \frac{2A^5}{15} + \frac{17A^7}{315} + \cdots $$Matrix tangent is especially useful in solving differential equations, transformation models, and advanced linear algebra applications.
Examples
Example 1
Compute the matrix tangent of:
$$ A = \begin{pmatrix}0.5 & 0 \\ 0 & 0.5\end{pmatrix} $$Since $A = 0.5I$, then:
$$\tan(A)=\tan(0.5)I$$So,
$$ \tan(A)=\tan(0.5)\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix} $$Example 2
Compute the matrix tangent of:
$$ B = \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix} $$Since $B^2 = 0$, the power series becomes:
$$\tan(B) = B$$Thus,
$$ \tan(B)=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix} $$Common Mistakes
- Applying tangent directly element-wise to a matrix. This is incorrect except for diagonal matrices.
- Forgetting that $\cos(A)$ must be invertible to compute $\tan(A)$.
- Ignoring higher-order terms when the matrix cannot be simplified using algebraic structure.
Practice Problems
Compute the matrix tangent for each matrix below.
Problem A (Angle values approximated as decimals)
1) $$ A_1 = \begin{pmatrix}0.524 & 0 \\ 0 & 0.524\end{pmatrix} $$
2) $$ A_2 = \begin{pmatrix}0.785 & 0 \\ 0 & 0.785\end{pmatrix} $$
Show Answers
$$\tan(A_1) = \tan(0.524)I$$ $$\tan(A_2) = \tan(0.785)I$$
Problem B (Pure numeric values)
3) $$ A_3 = \begin{pmatrix}0.5 & 1 \\ 0 & 0.5\end{pmatrix} $$
4) $$ A_4 = \begin{pmatrix}0 & 2 \\ 0 & 0\end{pmatrix} $$
Show Answers
For $A_3$: $\tan(A_3)$ requires series expansion.
For $A_4$: $$\tan(A_4)=A_4$$
Matrix tangent is often explored alongside matrix rank, SVD, and RREF.
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