Q, can a real symmetric matrix be diagonalized? Orthogonal matrices and symmetric matrices

If you can prove the following propositions, your problem will be solved immediately.

Let a be a nth-order real symmetric matrix, then we can find the nth-order orthogonal matrix t, such that (the inverse of t) at is a diagonal matrix.

Proof: When n=1, the conclusion is clearly true. Now it is proved that if the real symmetry matrix of order n-1 is true, then the real symmetry matrix of order n is also true.

Let be a eigenvalue of a (the nth order matrix must have n eigenvalues (counting repeats)) and let be an eigenvector of a (be a column vector). (of transpose)*a) of transpose=a=. Because the non-zero multiples of the eigenvector are still eigenvectors, as long as each element of is divided by , where the square of = (of the transpose)* makes the unit vector of (the so-called unit vector is the transpose of ( of the transpose) * = 1).

Obviously, there are an infinite number of unit vectors for all units, and it is obvious that enough column unit vectors can be found so that the inner product of their and 's is 0 and their inner product is equal to 0, because the sufficient condition of the orthogonal matrix is that the column (row) vectors are orthogonal and are unit vectors, and because for the opposing matrix, if ab=e, then ba=e, it can be thought that an orthogonal matrix q can be manually written for the first column, (the so-called orthogonal matrix is (the transpose of q)*q=q*(the transpose of q)=e). From the transpose of (( of transpose) * a = a = (the transpose of q) the first line of a is the transpose of (, so (the transpose of q) the first column of the first row of aq is ( of transpose) = , you can also launch (the transpose of q) the first column of aq is 0 except for the first row (as for why this is really inconvenient to type, the reader can calculate it himself, and remind him Let t be t is the yuan, tij*t+t..*t..

t..*t..+t..

t..If the corners of each item are not exactly the same, then these add up to 0). Because q is an orthogonal matrix, the transposition of ((inverse matrix of q) aq) = (transpose of q) (transpose of a) (transpose of inverse matrix of q) = (inverse matrix of q) aq, so (inverse matrix of q) aq is also a symmetry matrix, so it is 0 in the first row except for the first column, and a large piece of matrix left in the first column of the first row is still a symmetry matrix, so in the end, this process can be repeated into a diagonal matrix.

Certification. However, the orthogonal matrix must be an invertible matrix, which is invertibly equivalent to a full-rank matrix for the square matrix, and the rank does not change after multiplying a square matrix with a full-rank square, which proves that your real symmetric matrix must be similarly diagonal.

First question. Absolutely. But the other way around.

Matrices that can be diagonalized.

Not necessarily. Orthogonal matrices.

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1. The eigenvectors corresponding to the different eigenvalues of the real symmetry matrix a are orthogonal.

2. The eigenvalues of the real symmetry matrix a are all real equilibrium numbers, and the eigenvectors are all real vectors.

3. The nth-order real symmetry matrix a must be similar diagonal, and the elements on the similar diagonal family modulus are the eigenvalues of the matrix itself.

4. If a has a k-weight eigenvalue 0, there must be k linearly independent eigenvectors, or the rank r(0e-a) must be n-k, where e is the identity matrix.

5. The real symmetry matrix a must be orthogonally similar diagonal.

The conclusion that the Hermite matrix can be unitary diagonalized If a is a hermite matrix, take a unit eigenvector x and stretch it into a unitary matrix q=[x,*].

Then Q Haq has a chunked structure.

00 B uses the inductive hypothesis for B.

The eigenvalues of the real symmetric matrix are all real numbers, so the n-order matrix has n eigenvalues in the real number field, and the number of heavy shed grinds for each eigenvalue of the real symmetry matrix is the same as the number of irrelevant eigenvectors belonging to it.

In sexual algebra, the symmetry matrix is a square-beat wheel matrix in which the transpose matrix is equal to itself. In 1855, Emmett proved the special properties of the eigenroots of some matrix classes discovered by other mathematicians, such as the eigengen properties called Emmett matrices.

Later, Klebosch (1831-1872), Buckheim et al. proved the eigenroot properties of symmetry matrices. Taber (introduced the concept of matrix traces and gave some conclusions.

Matrices are a common tool in advanced algebra and are also commonly found in applied mathematics disciplines such as statistical analysis. In physics, matrices have applications in circuits, mechanics, optics, and quantum physics; In computer science, 3D animation also requires the use of matrices. The operation of matrices is an important problem in the field of numerical analysis.

Decomposing matrices into simple matrices can simplify the operation of matrices in theory and practical applications. For some matrices with wide applications and special forms, such as sparse matrices and quasi-diagonal matrices, there are specific fast operation algorithms.

The eigenvalues of the real symmetric matrix are all real numbers, so the n-order matrix has n eigenvalues (including multiplies) in the real number domain, and the repetition of each eigenvalue of the real symmetry matrix is the same as the number of irrelevant eigenvectors belonging to it, so that the n-order matrix has n irrelevant eigenvectors, so it can be diagonalized.

Conditions for judging whether a phalanx can be similar to diagonalization:

1) Sufficient and necessary conditions: The sufficient and necessary conditions for an to be similar to diagonalization are: an has n linear independent eigenvectors;

2) Another form of the sufficient and necessary condition: the sufficient and necessary condition for an to be similar to diagonalization is: the k-weight eigenvalue of an is satisfied with n-r( e-a)=k;

3) Sufficient conditions: if the n eigenvalues of an are different in pairs, then an must be similarly diagonalized;

4) Sufficient condition: If an is a real symmetry matrix, then an must be similarly diagonalized.