What are the important properties of diagonal matrices

Sufficient and necessary conditions for the n-order phalanx to be diagonalized.

Yes: There are n linearly independent eigenvectors in the order square.

Corollary: If this nth-order square matrix has n different eigenvalues.

Then there must be a similarity matrix in the matrix.

2.If there are duplicate eigenvalues in the order n square, the number of linearly independent eigenvectors for each eigenvalue is exactly equal to the weight of that eigenvalue.

Complex times. Now from the process of matrix diagonalization, let's talk about how this condition came about.

In the feature problem of matrices, the feature vector has a good property, i.e., aa= a.

Suppose there is a special case where a has n different eigenvalues i, i.e., aai = i*aiLet the matrix p=[a1a2

an] so that ap=a*[a1a2

an]=[a*a1a*a2

a*an]=[λ1*a12*a2

n*an]=p*b, where b is a diagonal array. b＝

nSince the eigenvectors corresponding to different eigenvalues are linearly independent, then p is an invertible matrix.

To put the above equation another way, it is.

a＝p*b*p-1

This is where A resemblance and Diagonal B are defined.

In this process, it is important for a person to be able to keratinization:

How is P composed? p consists of n linearly independent vectors, and the vectors come from the eigenvector space of a.

p to meet reversible. Under what circumstances is p reversible?

Matrices can be diagonalized.

The condition, in fact, is to ask under what circumstances is p reversible?

If a consists of n different eigenvalues, and one eigenvalue corresponds to one eigenvector, then it is easy to find n linearly independent eigenvectors and let them form p;

But if a has a certain is a heavy root.

This? For example, 3 is a triple root. We.

Know the corresponding characteristic equations.

3i a) x 0 does not necessarily have 3 linearly independent solutions. If 3 cannot find 3 linearly independent solutions, then A cannot be diagonalized because the p-matrix that makes A diagonalized does not exist.

Definition: An nth-order matrix in which all non-primary diagonal elements are all zero is called a diagonal matrix.

Properties: 1. The diagonal matrix is an nth-order square matrix.

2. The rank of the diagonal matrix is equal to the number of non-zero elements on the main diagonal.

3. The traces of the diagonal matrix are equal to the sum of the non-zero elements on the main diagonal.

4. The Jordan standard type of diagonal matrix is itself.

5. If none of the elements on the main diagonal of the diagonal matrix are zero, then the diagonal matrix is not singular, there is an inverse matrix, and the inverse matrix is also a diagonal matrix, and its main diagonal element is the reciprocal of the main diagonal element of the original diagonal matrix.

Matrix withDiagonal matrixA similar condition is: the smallest polynomial is noneHeavy roots, and the Gaelic circles do not intersect. Mathematically, a matrix is a two-dimensional data arranged vertically and horizontally**, which originally came from a square matrix composed of coefficients and constants of a system of equations.

This concept was first proposed by the 19th-century British mathematician John Kelly.

Mathematics (or maths, from Greek.

máthēma”；Often abbreviated as "math"), it is a discipline that studies concepts such as quantity, structure, change, space, and information.

A matrix with only non-zero elements on the diagonal is called a diagonal matrix, or if a square matrix has all but the elements on the main diagonal, all the elements are equal to zero.

The diagonal of a matrix has many properties, such as the invariance of the diagonal elements when doing transpose operations, and the sum of diagonal lines when similar transformations (called the traces of the matrix.

unchanged, etc. When studying matrices, many times it is necessary to extract the elements on the diagonal of the matrix to form a column vector, and sometimes it is necessary to construct a diagonal matrix with a vector of Shouchang.

Diagonal matrixif diagonal.

None of the elements on are 0, then this diagonal array is reversible.

its inverse matrix. It is also a diagonal matrix, and the elements on the diagonal are exactly the reciprocal of the elements on the diagonal of the corresponding original matrix.

It can be proved by the elementary transformation method of the inverse matrix, so the inverse matrix is as follows:

Diagonal matrixDiagonal matrix is a matrix in which all elements outside the main diagonal of a large grandson 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 in which the elements on the diagonal are equal 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.

Diagonal matrix:

A 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.

Quasi-diagonal matrix:

When a quasi-diagonal matrix is a matrix under the concept of a block matrix, that is, the matrix after the block is a diagonal matrix, it is called a quasi-diagonal matrix. The lower a is the chunked matrix:

Matrix A is a block matrix, and when the carry-on 2 in A is 0, it is a quasi-diagonal matrix, that is, matrix B is 0. Then the quasi-diagonal matrix is:

e1 = e3, of course e1 and e3 are not diagonal matrices.

A quasi-diagonal matrix is an example of the figure below

Diagonal matrix:

A diagonal matrix is a square matrix in which the main diagonal is generally not all 0 values, and the elements in the other positions are all 0s.

Meaning of diagonal matrix: a special case of symmetry matrix. Diagonal matrix (diagonalmatrix) is a special case of symmetric matrix, a specialized term in linear algebra.

We usually divide diagonal formations into positive diagonal formations and opposing diagonal formations. Diagonal matrixA diagonalmatrix is a matrix in which elements outside the main diagonal are believed to be 0. The element on the diagonal can be 0 or other values.

The formula is to make m=(ij) an n-order square, and all elements of m with equal two subscripts are called the diagonal elements of m, and the sequence (ii) and (1 i n) are called the main diagonal of m. Let m=(ij) be the square matrix of order n, all elements of m with equal two subscripts are called diagonal elements of m, and sequence (ii) (1 i n) is called the principal diagonal of m. A nth-order matrix in which all non-principal diagonal elements are equal to zero is called a diagonal matrix or diagonal matrix.

It is also often written as diag (a1, a2,..an) It is worth mentioning that the elements on the diagonal can be glide 0 or other values.

Therefore, a matrix of n rows and n columns = (a) if it satisfies the following properties: a then the matrix is a diagonal matrix. A matrix with all zeros on the diagonal is a special diagonal matrix, but it is generally referred to as a zero matrix.

The properties of the diagonal matrix are as follows:

The diagonal matrix is a square matrix, i.e., the number of rows and columns is equal.

None of the elements on the main diagonal of the diagonal matrix are zero, while the other elements are zero.

The inverse matrix of a diagonal matrix is also a diagonal matrix, and the elements on its main diagonal are the reciprocal of the elements on the main diagonal of the original matrix.

The determinant of a diagonal matrix is equal to the product of the elements on its main diagonal.

The eigenvalues of the diagonal matrix are equal to the elements on its main diagonal.

Mathematics is a general means for human beings to strictly describe and deduce the abstract structure and pattern of things, which can be applied to any problem in the real world, and all mathematical objects are artificially defined in nature.

In this sense, mathematics belongs to the formal sciences, not the natural sciences. Different mathematicians and philosophers have a range of opinions on the exact scope and definition of mathematics.

Mathematics is a universal means for humans to strictly describe the abstract structure and pattern of things, and it can be applied to any problem in the real world. In this sense, mathematics belongs to the formal sciences, not the natural sciences. All mathematical objects are inherently artificially defined, they do not exist in nature, but only in human minds and concepts.

Therefore, the correctness of mathematical propositions cannot be verified with the help of repeatable experiments, observations, or measurements, as in the natural sciences such as physics and chemistry, which aim at the study of natural phenomena, but can be directly proved by rigorous logical reasoning. Once a conclusion has been proved by logical reasoning, then the conclusion is correct.

The axiomatic method of number cluster ingenuity is essentially the direct application of the logical method in mathematics. In the axiomatic system, all propositions are connected to each other by rigorous logic.

Starting from the original concept that is directly adopted without definition, other derived concepts are gradually established through the means of logical definition; Starting from the axiom that is the premise without proof, with the help of logical deductive means, further conclusions, i.e., theorems; Then all the concepts and theorems are formed into a whole with internal logical connections, that is, the axiom system is formed.

Similar to the conditions of the diagonal matrix:1. A sufficient and necessary condition for the similarity of the square matrix to the diagonal matrix is that the square matrix has n linearly independent eigenvectors.

2. If there are several different eigenvectors in the matrix, these eigenvectors are linearly independent.

3. If the eigenvalues of the matrix are different from each other, it is similar to the diagonal matrix.

A 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 to be the simplest type of matrix.

The elements on the diagonal can be 0 or other values, and diagonally moment crypto-hail arrays with equal elements on the diagonal are called quantity matrices. 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 of the same order excitation diagonal matrix, and the result is still a diagonal matrix.