Calculation formula of inverse matrix of Hessian matrix

In mathematics, Hessian matrix (or Hessian) is a block matrix composed of the second partial derivatives of real valued functions whose independent variables are vectors. This function is as follows:

If all the second derivatives of F exist, then the Hessian matrix of F is as follows:
Where & nbsp;, i.e
Heisenberg matrix is applied to large-scale optimization problems solved by Newton method

Inverse matrix method
1) = (1 / | a |) × a * & nbsp;, where a ^ (- 1) represents the inverse matrix of matrix A, where | a | is the determinant of matrix A, and a * is the adjoint matrix of matrix A. & nbsp;

Another common method to find the inverse matrix: & nbsp;

(a | E) is transformed into (E | a ^ (- 1)). & nbsp;

Note: only row (column) operation is used for elementary change, not column (row) operation. E is the identity matrix;

In general, the following 11 cases will be encountered in calculation or judgment to judge the inverse matrix: & nbsp;

1 & nbsp; rank equals the number of rows & nbsp;

2 & nbsp; determinant is not 0 & nbsp;

3 & nbsp; row vectors (or column vectors) are linearly independent groups & nbsp;

4 & nbsp; there is a matrix whose product is the identity matrix;

5 & nbsp; as coefficients of linear equations, there is a unique solution;

6 & nbsp; full rank & nbsp;

7 & nbsp; can be transformed into identity matrix through elementary row transformation & nbsp;

8 & nbsp; adjoint matrix invertible
9 can be expressed as the product of elementary matrices;

Its transpose is reversible;

11 & nbsp; it multiplies another matrix to the left (right) with the same rank & nbsp;

The inverse matrix has the following properties: 1. The necessary and sufficient condition of invertibility of matrix A is that the determinant of a is not equal to 0;

2. An invertible matrix must be a square matrix;

If the matrix A is invertible, the inverse of a is unique;

4. Invertible matrices are also called nonsingular matrices and full rank matrices;

5 & nbsp; the product of two invertible matrices is still invertible;

The transpose matrix of invertible matrix is also invertible;

7 a matrix is invertible if and only if it is a full rank matrix;

Edit the solution in Matlab: inv (a) or a ^ - 1
For example: & nbsp;

　　>> a =

　　8 4 9

　　2 3 5

　　7 6 1

　　>> a^-1

　　ans =

　　0.1636 -0.3030 0.0424

　　-0.2000 0.3333 0.1333

　　0.0545 0.1212 -0.0970

　　>> inv(a)

　　ans =

　　0.1636 -0.3030 0.0424

　　-0.2000 0.3333 0.1333

　　0.0545 0.1212 -0.0970

The following is an explanation of the usage of inv in MATLAB;

Original text (from MATLAB & nbsp; help & nbsp; DOC) & nbsp;

　　In practice, it is seldom necessary to form the explicit inverse of a matrix. A frequent misuse of inv

　　arises when solving the system of linear equations Ax=B .

　　One way to solve this is with x = inv(A)*B.A better way, from both an execution time and numerical accuracy standpoint,is to use the matrix division operator x = A\b.

In fact, the exact value of matrix inverse is rarely needed. When solving the equation AX = B, we can use X & nbsp; = & nbsp; inv (a) * B, & nbsp;

But usually we don't need to find the inverse matrix of a when we solve this kind of linear equation. In MATLAB, the more accurate and faster method is to use left division - X & nbsp; = & nbsp; a / b. & nbsp;

In addition, LU decomposition method is faster, just write one more LU decomposition statement;

Speed can be estimated by TiC and TOC in MATLAB

2 times 2 inverse matrix algorithm formula verification process!
Why is there no inverse matrix when ad BC = 0?
This is the formula (1 / (AD BC)) * | D - B | - | - C a |?

When ad BC = 0, the determinant is 0, and the matrix is not full rank, so there is no inverse
The latter formula is based on the expansion formula of determinant

Finding the inverse of a square matrix
1 2 3
2 2 1
3 4 3
I calculated several times, and there is a gap with the correct answer

123 100221 010343 0011 2 3 1 0 00 -2 -5 -2 1 00 -2 -6 -3 0 11 2 3 1 0 00 -2 -5 -2 1 00 0 -1 -1 -1 11 0 -2 -1 1 00 -2 -5 -2 1 00 0 -1 -1 -1 11 0 -2 -1 1 00 -2 0 3 6 -50 0 1 1 1 -11 0 0 1 3 -20 1 0 -3/2...

What's the difference between the past tense and the past continuous tense?

The past continuous tense is often used to indicate the incompleteness of the action, while the past tense is usually used to indicate that the action has been completed
David wrote a letter to his father last night．
David wrote a letter to his father last night
David was writing a letter to his father last night．
David was writing to his father last night
2. The past continuous tense is used with always to express praise, exclamation and disgust, while the past tense is used with always to express frequent or repeated actions
He was always changing his mind．
He is always half hearted
He always went to work by bus last year．
He always went to work by bus last year
3. Generally speaking, the past continuous tense focuses on the length of action duration, while the past tense only indicates that something happened at a certain time in the past, focusing on facts
It was raining heavily last night．
It rained heavily last night
It rained heavily last night．
It rained heavily yesterday
4. When there are adverbials emphasizing the length of time, such as all night, the whole morning, it is better to use the past continuous tense
He was working all night last Monday．
He worked all night last Monday
5. In the adverbial clause of time, when is equivalent to while, there is little difference between the general past tense and the past continuous tense
I saw John when I walked／was walking to the station．
I saw John walking to the railway station
However, when the predicate verb in the when clause is the general past and its main sentence is the general past or the past, sometimes the time relationship will change
When we arrived,she was making some fresh coffee．
She was making coffee when we arrived
When we arrived,she made some fresh coffee．
When we arrived, she made coffee
Some verbs express state, feeling, psychology or emotion,
I had a lot of work to do yesterday

What is the principle of keyhole imaging?

A: pinhole imaging is formed by the linear propagation of light

What are the characteristics of capacitors and inductors?

Capacitor is the energy storage element in the circuit, which plays the role of filtering, bypass, coupling, tuning and waveform transformation in electronic circuit
As well as the generation of pulses and other functions
Inductor is also the energy storage element in the circuit, which plays the role of filtering, current limiting, tuning, oscillation, interference suppression and energy generation in the electronic circuit
The function of magnetic field

Using the principle of Marxist historical materialism, this paper analyzes and explains that the first priority of the scientific outlook on development is development, and the core is people-oriented
fast

1 and Marx's ruling party must attach great importance to the liberation and development of productive forces. We must always regard development as the first priority of the party in governing and rejuvenating the country, developing advanced productive forces, developing advanced culture and realizing the fundamental interests of the overwhelming majority of the people. Only then can we uphold the party's advanced nature,.2, social justice, and realize superiority and realize the prosperity of the country and the people.

Is my favorite food and drink singular or plural

The plural, because it is two kinds, eat and drink

Calculate the surface area and volume of a cylinder with height of 12cm and diameter of 6cm
Step by step, I don't understand the process,

Surface area of cylinder = side area + two bottom areas
=3.14×6×12＋3.14×(6÷2)²×2
=226.08＋56.52
=282.6(cm²)
Volume of cylinder = bottom area × height
=3.14×(6÷2)²×12
=28.26×12
=339.12(cm³)

What is the greatest common factor of 15 and 20 The least common multiple is ．

15 = 3 × 5, 20 = 2 × 2 × 5, so the greatest common factor of 15 and 20 is 5, and the least common multiple is 2 × 2 × 3 × 5 = 60

Calculation formula of inverse matrix of Hessian matrix