Difference between revisions of "Identity matrix"
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| − | < | + | In [[linear algebra]], the square '''identity matrix''' is a <math>n \times n</math> matrix with <math>1</math>s in its [[main diagonal]] and <math>0</math>s in every other entry. It is usually denoted <math>I_n</math>. |
| − | + | <cmath>I_n = \begin{pmatrix}1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\end{pmatrix}.</cmath> | |
| − | + | The corresponding linear map is the identity map. For any <math>n \times n</math> matrix <math>A</math>, we have <math>AI_n = I_nA = A</math>. The [[matrix inverse|inverse]] <math>A^{-1}</math> of <math>A</math> is the unique matrix such that <math>AA^{-1} = A^{-1}A = I_n</math>. | |
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| − | + | The [[determinant]] of <math>I_n</math> is <math>1</math>. <math>I_n</math> has only one [[eigenvalue]] <math>1</math>, occurring with multiplicity <math>n</math>. Hence, any <math>n \times n</math> matrix is in the corresponding eigenspace. The [[characteristic polynomial]] of <math>I_n</math> is <math>P_{I}(t) = (t-1)^n</math>, and the [[minimal polynomial]] is <math>t - 1</math>. | |
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| − | + | [[Category:Linear algebra]] | |
Latest revision as of 08:57, 20 February 2025
In linear algebra, the square identity matrix is a 
matrix with 
s in its main diagonal and 
s in every other entry. It is usually denoted 
.
![\[I_n = \begin{pmatrix}1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0\\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1\end{pmatrix}.\]](http://latex.artofproblemsolving.com/1/d/7/1d7167bc6b72d17dc300a2544c533133c7fa6eb4.png)
The corresponding linear map is the identity map. For any matrix 
, we have 
. The inverse 
of is the unique matrix such that 
.
The determinant of is . has only one eigenvalue , occurring with multiplicity 
. Hence, any matrix is in the corresponding eigenspace. The characteristic polynomial of is 
, and the minimal polynomial is 
.
