Matrix Operation

pic. 1

Let's think about possible conditions of matrix operation

by reading pic. 1.

Example

pic. 2

pic. 3

There are n x n matrices, A and B.

Every matrix has rows and columns like pic. 3.

Let's operate matrices.

pic. 4

We can operate like pic. 4.

Example

pic. 5

Examples of Matrix Operation

Vector in Coordinate System

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Eigenvalue in Matrix

Diagonalizable

Total differential

Unit Vector

pic. 1

Let formulas be supposed like that pic. 1.

pic. 2

We can express the vector v like pic. 2.

pic. 3

We can find relationship 

between total differentials in Cartesian and general coordinate system

in pic. 2 and pic. 3.

Fibonacci Number

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Diagonalizable

Gauss - Jordan Elimination

Gauss - Jordan Elimination

If you solve simultaneous equations or find inverse matrix,

you can use Gauss-Jordan Elimination.

Diagonalizable

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Properties of Determinant

Eigenvalue in Matrix

Eigenvalue in Matrix

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Determinant A where Ax=O

Determinant A where Ax=O

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Properties of Determinant

Properties of Determinant

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Determinant of Matrix

1. det [cA] = c^n det A

2. DETERMINANT of A MATRIX 

that is multiplied constants at a column of a certain matrix

 and added same column of another matrix 

3. determinant of exchanged 2 columns of matrix

If a_ik=a_il, where i=1,2, ... ,n, 

that means col_k [A] = col_l [A],

det A = det C.

So, det A = - det A.

Then, det A = 0.

Inverse Matrix

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Identity Matrix

Determinant of Matrix

Identity Matrix

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Determinant of Matrix

Determinant of Matrix

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Permutation, Symmetric Group and Signature

Cyclic Permutation

Matrix Operation