Inverse functions and the identity function - Revision history

Khanh at 02:09, 15 November 2021

2021-11-15T02:09:36Z

Khanh at 05:52, 3 October 2021

2021-10-03T05:52:49Z

Khanh at 05:50, 3 October 2021

2021-10-03T05:50:57Z

Show changes

Khanh: Created page with "* [https://www.youtube.com/watch?v=kWorj5BBy9k Inverse Matrices and Their Properties], video by Professor Dave Explains * [https://www.khanacademy.org/math/algebra-home/alg-ma..."

2021-09-27T02:12:06Z

Created page with "* [https://www.youtube.com/watch?v=kWorj5BBy9k Inverse Matrices and Their Properties], video by Professor Dave Explains * [https://www.khanacademy.org/math/algebra-home/alg-ma..."

New page

* [https://www.youtube.com/watch?v=kWorj5BBy9k Inverse Matrices and Their Properties], video by Professor Dave Explains
* [https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-intro-to-matrix-inverses/v/inverse-matrix-part-1 Introduction to Matrix Inverses], Khan Academy

@@ Line 323: / Line 323: @@
 * [https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-intro-to-matrix-inverses/v/inverse-matrix-part-1 Introduction to Matrix Inverses], Khan Academy
-==References==
+== Licensing ==
-# "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-09-08.
+Content obtained and/or adapted from:
-# "Invertible Matrices". www.sosmath.com. Retrieved 2020-09-08.
+* [https://en.wikipedia.org/wiki/Invertible_matrix Invertible matrix, Wikipedia] under a CC BY-SA license

@@ Line 246: / Line 246: @@
 === Reciprocal basis vectors method ===
-Given an <math>n \times n</math> square matrix <math>\mathbf{X} = \left[ x^{ij} \right] </math>, <math> 1 \leq i,j \leq n </math>, with <math>n</math> rows interpreted as <math>n</math> vectors <math>\mathbf{x}_{i} = x^{ij} \mathbf{e}_{j}</math> ([[Einstein summation]] assumed) where the <math>\mathbf{e}_{j}</math> are a standard orthonormal basis of Euclidean space <math>\mathbb{R}^{n}</math> (<math>\mathbf{e}_{i} = \mathbf{e}^{i}, \mathbf{e}_{i} \cdot \mathbf{e}^{j} = \delta_i^j</math>), then using Clifford algebra (or Geometric Algebra) we compute the reciprocal (sometimes called dual) column vectors <math>\mathbf{x}^{i} = x_{ji} \mathbf{e}^{j} = (-1)^{i-1} (\mathbf{x}_{1} \wedge\cdots\wedge ()_{i} \wedge\cdots\wedge\mathbf{x}_{n}) \cdot (\mathbf{x}_{1} \wedge\ \mathbf{x}_{2} \wedge\cdots\wedge\mathbf{x}_{n})^{-1} </math> as the columns of the inverse matrix <math>\mathbf{X}^{-1} = [x_{ji}]</math>. Note that, the place "<math>()_{i}</math>" indicates that "<math>\mathbf{x}_{i}</math>" is removed from that place in the above expression for <math>\mathbf{x}^{i}</math>. We then have <math>\mathbf{X}\mathbf{X}^{-1} = \left[ \mathbf{x}_{i} \cdot \mathbf{x}^{j} \right] = \left[ \delta_{i}^{j} \right] = \mathbf{I}_{n} </math>, where <math>\delta_{i}^{j}</math> is the [[Kronecker delta]]. We also have <math>\mathbf{X}^{-1}\mathbf{X} = \left[\left(\mathbf{e}_{i}\cdot\mathbf{x}^{k}\right)\left(\mathbf{e}^{j}\cdot\mathbf{x}_{k}\right)\right] = \left[\mathbf{e}_{i}\cdot\mathbf{e}^{j}\right] = \left[\delta_{i}^{j}\right] = \mathbf{I}_{n}</math>, as required. If the vectors <math>\mathbf{x}_{i}</math> are not linearly independent, then <math>(\mathbf{x}_{1} \wedge \mathbf{x}_{2} \wedge\cdots\wedge\mathbf{x}_{n}) = 0</math> and the matrix <math>\mathbf{X}</math> is not invertible (has no inverse).
+Given an <math>n \times n</math> square matrix <math>\mathbf{X} = \left[ x^{ij} \right] </math>, <math> 1 \leq i,j \leq n </math>, with <math>n</math> rows interpreted as <math>n</math> vectors <math>\mathbf{x}_{i} = x^{ij} \mathbf{e}_{j}</math> (Einstein summation assumed) where the <math>\mathbf{e}_{j}</math> are a standard orthonormal basis of Euclidean space <math>\mathbb{R}^{n}</math> (<math>\mathbf{e}_{i} = \mathbf{e}^{i}, \mathbf{e}_{i} \cdot \mathbf{e}^{j} = \delta_i^j</math>), then using Clifford algebra (or Geometric Algebra) we compute the reciprocal (sometimes called dual) column vectors <math>\mathbf{x}^{i} = x_{ji} \mathbf{e}^{j} = (-1)^{i-1} (\mathbf{x}_{1} \wedge\cdots\wedge ()_{i} \wedge\cdots\wedge\mathbf{x}_{n}) \cdot (\mathbf{x}_{1} \wedge\ \mathbf{x}_{2} \wedge\cdots\wedge\mathbf{x}_{n})^{-1} </math> as the columns of the inverse matrix <math>\mathbf{X}^{-1} = [x_{ji}]</math>. Note that, the place "<math>()_{i}</math>" indicates that "<math>\mathbf{x}_{i}</math>" is removed from that place in the above expression for <math>\mathbf{x}^{i}</math>. We then have <math>\mathbf{X}\mathbf{X}^{-1} = \left[ \mathbf{x}_{i} \cdot \mathbf{x}^{j} \right] = \left[ \delta_{i}^{j} \right] = \mathbf{I}_{n} </math>, where <math>\delta_{i}^{j}</math> is the Kronecker delta. We also have <math>\mathbf{X}^{-1}\mathbf{X} = \left[\left(\mathbf{e}_{i}\cdot\mathbf{x}^{k}\right)\left(\mathbf{e}^{j}\cdot\mathbf{x}_{k}\right)\right] = \left[\mathbf{e}_{i}\cdot\mathbf{e}^{j}\right] = \left[\delta_{i}^{j}\right] = \mathbf{I}_{n}</math>, as required. If the vectors <math>\mathbf{x}_{i}</math> are not linearly independent, then <math>(\mathbf{x}_{1} \wedge \mathbf{x}_{2} \wedge\cdots\wedge\mathbf{x}_{n}) = 0</math> and the matrix <math>\mathbf{X}</math> is not invertible (has no inverse).
 == Derivative of the matrix inverse ==