Matrix Arctangent Function: Definition, Examples, and Practice

This article explains the matrix arctangent function, gives formulas and worked examples, summarizes common mistakes, and provides practice problems with collapsible answers. Suitable as companion content for a Matrix Arctangent Calculator.

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Definition and Formula

The matrix arctangent function, denoted as $\arctan(A)$, generalizes the scalar inverse tangent to square matrices. It can be defined using the matrix tangent function:

$$ \arctan(A) = \tan^{-1}(A) $$

More practically, the matrix arctangent is computed using its Taylor power series expansion:

$$ \arctan(A) = A - \frac{A^3}{3} + \frac{A^5}{5} - \frac{A^7}{7} + \cdots $$

The series converges when $\|A\| < 1$ and is widely used in numerical linear algebra, signal processing, and control systems.

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Examples

Example 1

Compute the matrix arctangent of:

$$ A = \begin{pmatrix}0.2 & 0 \\ 0 & 0.2\end{pmatrix} $$

Since $A = 0.2I$, then:

$$ \arctan(A)=\arctan(0.2)I $$

Example 2

Compute $\arctan(B)$ for:

$$ B = \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix} $$

Because $B^2 = 0$, we can simplify the series:

$$ \arctan(B) = B $$

Thus,

$$ \arctan(B)=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix} $$

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Common Mistakes

• Applying $\arctan$ to each matrix element directly. This is only valid for diagonal matrices.
• Incorrectly truncating the series for matrices where $A^2 \neq 0$.
• Forgetting that $\arctan(A)$ requires convergence conditions for exact results.

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Practice Problems

Compute the matrix arctangent for each matrix below.

Problem A (Diagonal matrices)

1) $$ A_1 = \begin{pmatrix}0.1 & 0 \\ 0 & 0.1\end{pmatrix} $$

2) $$ A_2 = \begin{pmatrix}0.3 & 0 \\ 0 & 0.3\end{pmatrix} $$

Show Answers

$$\arctan(A_1) = \arctan(0.1)I$$ $$\arctan(A_2) = \arctan(0.3)I$$

Problem B (Non-diagonal matrices)

3) $$ A_3 = \begin{pmatrix}0 & 0.5 \\ 0 & 0\end{pmatrix} $$

4) $$ A_4 = \begin{pmatrix}0.4 & 1 \\ 0 & 0.4\end{pmatrix} $$

Show Answers

For $A_3$: $$\arctan(A_3)=A_3$$

For $A_4$: $$\arctan(A_4) = A_4 - \frac{A_4^3}{3} + \cdots$$ (series expansion required)

Matrix arctangent is also related to matrix rank, SVD, and RREF.

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