watch: LA - 高山 13 | Application of Inverse Matrix

Inverse of diagonal matrix

A diagonal matrix with the given values: 2,3,4 as follows.

$$ 𝐀 = \text{diag\{2, 3, 4\}} = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{bmatrix} $$

The augmented matrix is constructed and perform elementary row operations as:

$$ 𝐀|𝐈 = \begin{bmatrix} 2 & 0 & 0 & | 1 & 0 & 0 \\ 0 & 3 & 0 & | 0 & 1 & 0 \\ 0 & 0 & 4 & | 0 & 0 & 1 \end{bmatrix} \overset{1/2r1,1/3r2, 1/4r3}{⇢} \begin{bmatrix} 1 & 0 & 0 & | 1/2 & 0 & 0 \\ 0 & 1 & 0 & | 0 & 1/3 & 0 \\ 0 & 0 & 1 & | 0 & 0 & 1/4 \end{bmatrix} $$

From the results, the inverse matrix 𝐀⁻¹ of a diagonal matrix is also a diagonal matrix. And its elements on the main diagonal are the reciprocal of elements on the original diagonal, where the element cannot be 0.

$$ 𝐀⁻¹ = \begin{bmatrix} 1/2 & 0 & 0 \\ 0 & 1/3 & 0 \\ 0 & 0 & 1/4 \end{bmatrix} = \text{diag\{1/2, 1/3, 1/4\}} $$

Inverse of a scalar matrix

If all elements on the diagonal are the same, the matrix is a schalar matrix. 𝚲 = diag{λ, …, λ}, λ≠0.

Its inverse matrix is a scalar matrix where the elements on the diagonal are all 1/λ, 𝚲⁻¹ = diag{λ⁻¹, …, λ⁻¹}, λ≠0.

If λ = 1, the scalar matrix is identity matrix 𝐈. So the inverse of a identity matrix is itself: 𝐈⁻¹ = 𝐈.

Inverse of the elementary matrix

The elementary matrix is definitely invertible. So their inverse matrices must exist.

Inverse of a row-switching matrix

For a row-switching matrix (permutation matrix) switching the first 2 rows of the identity matrix:

$$ 𝐅₁ = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Write the augmented matrix: $$ 𝐅₁|𝐈 = \begin{bmatrix} 0 & 1 & 0 &| 1 & 0 &0\\ 1 & 0 & 0 &| 0 & 1 &0\\ 0 & 0 & 1 &| 0 & 0 &1 \end{bmatrix} $$

The inverse matrix can be obtained by switching the first two rows and letting the left part become a identity matrix. Then the inverse matrix is the right part:

$$ 𝐅₁⁻¹ = \begin{bmatrix} 0 & 1 &0\\ 1 & 0 &0\\ 0 & 0 &1 \end{bmatrix} $$

It indicates that the inverse of a row-switching matrix is itself. (Just like the identity matrix, its inverse is itself.)

Inverse of a row-multiplying matrix

For a row-multiplying matrix scaling the second rows by 2 times:

$$ 𝐅₂ = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Therefore, its inverse is obtained by multiplying 1/2 onto the 2nd row of the identity matrix.

$$ 𝐅₂⁻¹ = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1/2 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

That means the inverse of a row-multiplying matrix will still be a row-multiplying matrix. But the scale factor becomes the reciprocal.

Inverse of a row-addition matrix

For a row-addition matrix: the second row is added by 2-times first row. $$ 𝐅₃ = \begin{bmatrix} 1 & 0 & 0 \\ 2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Its inverse is obtained by lettingthe 2nd row in the identity matrix add the -2-times of the first row:

$$ 𝐅₃⁻¹ = \begin{bmatrix} 1 & 0 & 0 \\ -2 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$

Thus, its inverse is just changing the scale factor to its opposite.

In summary, the inverse matrices of elementry matrices are the same type.

The inverse of a row-switching matrix is itself.
The inverse of a row-multiplication matrix is changing the scale factor to its reciprocal.
The inverse of a row-addition matrix is changing the scale factor to its opposite.

Inverse of second-order matrix

The number of rows and columns of a second-order matrix are both 2.