How to calculate the characteristic equation of matrix A

Because of the characteristic equations.

Equals: |λe-a|==0 Calculation Process:

Therefore, ( -2) (1)=0 is obtained, and the eigenvalue is -1,2 (which is the double eigenroot).

Let the eigenvalues of this matrix a be

Rule. a-λe|=

1 -3 -3 - Line 1 minus line 3 multiplied by 1 -3 -3 - By column 1.

The eigenvalue = -1 is a triple eigenvalue.

That determinant is calculated directly, and after simplification, it's the thing you asked. A determinant does not change the original value by adding or subtracting rows and columns, and then evaluates quickly through the determinant, but the value between the rows and columns is uncertain, so it is best not to use the rows and columns with the input to transform.

<> "Let the matrix a be an nth-order matrix, the eigenvalue is , and the eigenvector is x, then there is: ax = x to shift the term to obtain: (a- i)x = 0, where i is the nth-order identity matrix.

The solution x of this equation cannot be a zero vector, i.e., the matrix (a-i) has an rank n-1. Therefore, the eigenvalue is satisfying the equation |a-λi|The root of =0, i.e., the characteristic equation of matrix a, is: |a-λi|=0 solves the root of the eigenequation 1, 2,..

n, we can get all the eigenvalues of matrix a.

All the eigenvalues of matrix a are: 1=0, 2=3, 3=-6.

Calculation process: a-e|=0 because a=

*6)*(3), so the eigenvalue is , -6.

a*|Equal to 4.|a^2-2a+e|Equal to 0.

Solution: Since the eigenvalues of matrix a are 1=-1, 2=1, 3=2, then |a|=λ1*λ2*λ3=-1*1*2=-2。

According to |a*| a|(n-1), you can find |a*|=a|^2 = 2)^2 = 4。

At the same time, according to the properties of the eigenvalues of the matrix, the eigenvalues of a2-2a+e can be obtained as 1, 2, and 3.

then 1=( 1) 2-2 1+1=4, 1=( 2) 2-2 2+1=0, 3=( 3) 2-2 3+1=1, then |a^2-2a+e|=η1*η2*η3=4*0*1=0

i.e. |a*|Equal to 4.|a^2-2a+e|Equal to 0.

1. Let x be the eigenvector of matrix a, and calculate ax first;

2. It is found that the resulting vector is a multiple of x;

3. Calculate the multiple, which is the required eigenvalue.

The method for finding all the eigenvalues and eigenvectors of a matrix is as follows:

Step 1: Compute the feature polynomial;

Step 2: Find all the roots of the eigenequation, that is, all the eigenvalues;

Step 3: For each eigenvalue, find a basic solution system of homogeneous linear equations, and then find all the eigenvectors that belong to the eigenvalues.

The characteristic equation of the matrix is:

a * x = lamda * x

What can be seen in this equation? The matrix can actually be seen as a transformation, and the left side of the equation is just a change of vector x to another position; On the right, the vector x is made into a stretch letter, and the stretch is lamda. Then its significance is obvious, expressing that one of the properties of matrix A is that this matrix can lengthen (or shorten) vector x by a factor of lamda, and that's it.

Given any given matrix a, it is not possible to elongate (shorten) all x's. Any vector that can be elongated (shortened) by a is called an eigenvector; The elongated (shortened) amount is the eigenvalue corresponding to the eigenvalue of the eigenvector.

It is worth noting that the eigenvectors we are talking about are a class of vectors, because any eigenvector multiplied by a scalar must also satisfy the above equation, of course, both vectors can be regarded as the same eigenlandslide vector, and they both correspond to the same eigenvalue.

If the eigenvalues are negative, it means that the matrix not only stretches the vector as long (shortened), but also points the vector in the opposite direction.

The solution process is as follows:

1) The rank of the inverse matrix is obtained from the rank of matrix a.

2) According to the solution of the inverse matrix, the adjoint matrix expression is obtained.

3) Columnar solution defined by eigenvalues.

a-λe|=0, eigenvalue, is the subtraction of the main diagonal element, while the diagonal matrix, the eigenvalue and the diagonal element are equal, just satisfying the |a-λe|=0 diagonal matrix is a matrix in which all elements outside the main diagonal are 0, often written as diag(a1,a2,..an) 。

Diagonal matrices can be considered the simplest of the matrices, and it is worth mentioning that the elements on the diagonal can be 0 or other values, and the diagonal matrix of the stool with equal elements on the diagonal is called the quantity matrix; A diagonal matrix with all elements on the diagonal being 1 is called an identity matrix.

The operation of the diagonal matrix includes the sum operation, the difference operation, the number multiplication operation, and the product operation of the diagonal matrix of the same order, and the result is still the diagonal matrix.

To find the eigenvector, let a be the nth order matrix, and write ( e-a)x=0 according to the relation ax = x, and then write the eigenpolynomial e-a|=0, we can find that matrix A has n eigenvalues (including heavy eigenvalues).

Substituting the eigenvalue i into the original eigenpolynomial, the equation (ie-a) x=0 is solved, and the solution vector x is the eigenvector of the corresponding eigenvalue i.

First of all, write the flutter column |λe-a|By definition, the determinant is the sum of the products of terms that are not in the same row and can only be obtained by (n-1) by taking the product of the elements on the diagonal (-a11) (a22)...ann), so the n-1 coefficients of the eigenpolynomial are -(a11+a22+..ann), while the feature polynomial = (1)(2).

n), n-1 subterm coefficient is -( 1+ 2+.n), so a11+a22+.ann=λ1+λ2+..

n。From this it can be proved that the sum of the eigenvalues is equal to the sum of the elements on the main diagonal of the matrix.

By definition, ax=cx: a is the matrix, c is the eigenvalue, and x is the eigenvector. The matrix a is multiplied by x, and the vector x is transformed (rotated or stretched) (which is a linear transformation), and the effect of this transformation is the constant c multiplied by vector x (i.e., only stretched).

In general, finding eigenvalues and eigenvectors is to find out which vectors (of course, eigenvectors) can be stretched only by the matrix, and to what extent (eigenvalue size) can be stretched. Extended Information: Principles of Numerical Calculation:

In practice, the eigenvalues of large matrices cannot be calculated by eigenpolynomials, which are themselves quite resource-intensive, and the exact "symbolic" roots are difficult to calculate and express for higher-order polynomials: Abel Rufenidine's ingenious theorem shows that the roots of higher-order (5th or more modular) polynomials cannot be simply expressed by n-power roots. There are effective algorithms for estimating the root of polynomials, but small errors in eigenvalues can lead to large errors in eigenvectors.

The general algorithm for finding the zero point of the eigenpolynomial, i.e., the eigenvalue, is an iterative method. The simplest method is the exponential method: take a backup random vector v and compute a series of unit vectors.

First of all, write the flutter column |λe-a|By definition, the determinant is the sum of the products of terms that are not in the same row and can only be obtained by (n-1) by taking the product of the elements on the diagonal (-a11) (a22)...ann), so the n-1 coefficients of the eigenpolynomial are -(a11+a22+..ann), while the feature polynomial = (1)(2).

n), n-1 subterm coefficient is -( 1+ 2+.n), so a11+a22+.ann=λ1+λ2+..

n。From this it can be proved that the sum of the eigenvalues is equal to the sum of the elements on the main diagonal of the matrix.