Review of Linear Algebra

# Review of Linear Algebra

Donny
October 20, 2021
October 21, 2021
336
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## Matrix

Basic Operations

• Matrix Product (Dot Product)
• Matrix Division

Matrix Operations

Matrix Factorization (?)

Others:

• Unitary Matrix
A matrix $$U$$ that satisfies
• Minor Matrix
For square matrix A,  $$adj(A) = C^{T}$$ , $$C = (-1)^{i+j} M_{i,j}$$ , where $$M_{i,j}$$  is the (i, j) minor of A.
• Conjugate Transpose
$$(A^{*})_{i,j} = \overline{A_{j,i}}$$ , for complex number $$c = a+ib$$ , $$\overline{c} = a-ib$$
• Orthogonal Matrix
A matrix that satisfy $$A^{T} = A^{-1}$$

### Matrix Product

Matrix product is produced by taking i-th row of the first matrix and j-th column of the second matrix and calculate dot product of the two vectors as the (i, j) element in the result matrix.

Given

$$A = \begin{bmatrix} 1 & 2 & 1 \\ 3 & 4 & 2 \ \end{bmatrix} \\ B = \begin{bmatrix} 1 & 2 \\ 2 & 1 \\ 1 & 2 \ \end{bmatrix} \\$$

then,

$$AB = \begin{bmatrix} 6 & 6 \\ 13 & 14 \end{bmatrix}$$

### Matrix Division

Matrix Division makes use of the following property:

$$AX = B \\ A^{-1}AX = A^{-1}B \\ X = A^{-1}B$$

where  $$A^{-1}$$  is the inversion of A. Therefore A must be invertible (see Invertible Matrix, in short a matrix is invertible if it has full rank).

### Matrix Inversion

An n-by-n Matrix A is invertible if there exists an n-by-n matrix B such that $$BA = BA = I_n$$ . And B is called the inverse of A.

Only square matrices can be invertible. Non-square matrix may have left-inverse or right-inverse.

A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero.

Given

$$A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix}$$

The inverse of A, $$A^{-1}$$ is calculated by

$$[A | I ] = \begin{bmatrix} 1 & 3 & 3 & | & 1 & 0 & 0 \\ 1 & 4 & 3 & | & 0 & 1 & 0 \\ 1 & 3 & 4 & | & 0 & 1 & 0 \end{bmatrix}$$

Transform it using Gaussian Elimination (aka. row reduction) so that the left side becomes an identity matrix,

$$[I | A^{-1}] = \begin{bmatrix} 1 & 0 & 0 & | & 7 & -3 & -3 \\ 0 & 1 & 0 & | & -1 & 1 & 0 \\ 0 & 0 & 1 & | & -1 & 0 & 1 \\ \end{bmatrix}$$

Therefore, $$A^{-1}$$ is

$$A^{-1} = \begin{bmatrix} 7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1 \\ \end{bmatrix}$$

The inverse can also be calculated as:

$$A^{-1} = \frac{1}{det(A)} adj(A)$$