It is proved that the orthogonal matrix whose eigenvalues are all real numbers is a symmetric matrix Mayday: how to prove it. I'm almost broken Don't talk nonsense. This question is the subject of postgraduate entrance examination for Advanced Algebra over the years. It must be correct. Otherwise, counter examples will be given

This is a false proposition. If an orthogonal matrix whose eigenvalues are all real numbers is a symmetric matrix, a condition must be attached: eigenvalue = 1
If you can't figure it out, ask me again. I pushed it!

Is the orthogonal matrix of a real symmetric matrix unique

You mean p ^ - 1AP = orthogonal matrix P in diagonal matrix
It's not unique
The column vector of P comes from the basic solution system of corresponding homogeneous linear equations
The fundamental solution system is not unique
So p is not unique

Orthogonal matrix and symmetric matrix
I once saw it in a book, saying: "orthogonal matrix must be symmetric matrix." but I can't figure it out how to prove it. I hope someone can give me a detailed proof process!

Since it's axiom, you should not go to deliberate verification. The predecessors have been studying hard. I really want to know. Ask the teacher, and the teacher should be happy to tell you the verification process

Let a and B be real symmetric matrices of order n, then there exists an orthogonal matrix P such that p'ap and p'bp are diagonal matrices if and only if AB = ba

Not only that, but also A1 , an are similar to diagonal matrices, AIAJ = ajai. (I ≠ J)
Let p ^ (- 1) AIP I = 1 (this is 1978
A question in the second entrance examination of master degree students majoring in algebra in Wuhan University
Please refer to: Dictionary of algebra, edited by fan Yun, central China Normal University Press, 937 questions, 940 questions
P431－P432.

AB is a real symmetric matrix of order n, a is positive definite. It is proved that there exists a real invertible matrix P of order n such that p'ap and p'bp are diagonal matrices (p 'is transpose matrix)

Because a is positive
So there is an invertible matrix C such that c'ac = E
For a real symmetric matrix c'bc, there exists an orthogonal matrix D such that d '(c'bc) d is a diagonal matrix
D '(c'ac) d = d'd = e is also a diagonal matrix
Therefore, if P = CD, the requirement is satisfied

Let n-order matrix a be symmetric positive definite and n-order matrix B be symmetric. It is proved that there exists a congruent transformation matrix P such that p'ap and p'bp are diagonal matrices

There is an invertible matrix m such that
M'AM=E
In this case, m'bm is still symmetric, so there is an orthogonal matrix Q such that
Q'M'BMQ=D
D is a diagonal matrix
Let P = MQ

Let a =, find an orthogonal matrix P, where p ^ (- 1) AP is a diagonal matrix
A=2 0 0
0 0 1
0 1 0

λE-A=
λ-2 0 0
0 λ -1
0 -1 λ
|λE-A|=λ^2(λ-2)-(λ-2)=(λ+1)(λ-1)(λ-2)
So the eigenvalues of matrix A are λ 1 = - 1, λ 2 = 1, λ 3 = 2
When λ 1 = - 1, the basic solution system of the equation system (λ e-A) x = 0 is X1 * = (0, - 1,1) ^ t
So the eigenvector corresponding to the eigenvalue λ 1 = - 1 is X1 * = (0, - 1,1) ^ t, and the unit is X1 = (0, - 2 / 2, √ 2 / 2) ^ t
When λ 2 = 1, the basic solution system of the equation system (λ e-A) x = 0 is x2 * = (0,1,1) ^ t
So the eigenvector corresponding to the eigenvalue λ 2 = 1 is x2 * = (0,1,1) ^ t, and the unit is x2 = (0, √ 2 / 2, √ 2 / 2) ^ t
When λ 3 = 2, the basic solution system of the equation system (λ e-A) x = 0 is X3 * = (1,0,0) ^ t
So the eigenvector corresponding to the eigenvalue λ 3 = 2 is X3 * = (1,0,0) ^ t, and the unit is X3 = (1,0,0) ^ t
Let P = (x1, X2, x3)=
0 0 1
-√2/2 √2/2 0
√2/2 √2/2 0
So the matrix P is exactly what we want, so that P ^ (- 1) AP = (- 100; 0 10; 0 02) is a diagonal matrix

Let a and B be symmetric matrices of order n, and prove that AB is symmetric if and only if AB = ba

Sufficiency: because AB = Ba, so (AB) '= b'a' = Ba = AB, so AB is a symmetric matrix
Necessity: because AB is a symmetric matrix, ab = (AB) '= b'a' = ba

Let AB be symmetric matrices of order n. It is proved that AB is symmetric if and only if a and B are commutative

It is proved that a '= a, B' = B because a and B are symmetric matrices of order n
AB is a symmetric matrix
(AB)' = AB
B'A' = AB
BA=AB
A and B are interchangeable

It is proved that if a and B are symmetric matrices of order n, then AB is symmetric if and only if a and B are commutative

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