# inequality – Module of eigenvalues ​​of the Hermitian matrix under certain conditions

Leave $$H$$ be a $$n times n$$ Hermitian matrix that satisfies

$$sum_i ^ n | H_ {ij} | ^ 2 leq 1, ~~ sum_j ^ n | H_ {ij} | ^ 2 leq 1.$$
This means that the norm of each column or row vector of $$H$$ is less than 1

So my question is: "Does this condition imply the maximum module of its own values? $$| lambda_i |$$ be less than some constant? "

For the similar case, the module of all the eigenvalues ​​of the stochastic matrix $$P$$ what satisfies $$sum_i P_ {ij} = 1$$ is less than 1

## 2 for 2 case

First I tried $$2 times 2$$ matrix.
The general $$2 times 2$$ Hermitian matrix $$M$$ It can be written as

$$M = begin {bmatrix} a + d & b + ic \ b-id and a-d end {bmatrix}$$
where $$a, b, c, d$$ they are real numbers
So the previous condition implies

$$(a + d) ^ 2 + b ^ 2 + c ^ 2 leq 1 \ (a-d) ^ 2 + b ^ 2 + c ^ 2 leq 1 \$$

In addition, the two proper values ​​for $$M$$ is

$$lambda { pm = a pm sqrt {b ^ 2 + c ^ 2 + d ^ 2}$$
As $$± a$$ gives the same maximum module (if $$a> 0$$, $$lambda _ {+}$$ will give the maximum module, and vice versa), I only consider the maximum value of $$lambda _ {+}$$ with $$a> 0$$.
Squared both sides, we get

$$( lambda_ + -a) ^ 2 = b ^ 2 + c ^ 2 + d ^ 2$$
Using the two inequalities, I get

$$( lambda_ + -a) ^ 2 = b ^ 2 + c ^ 2 + d ^ 2 leq 1-a ^ 2-2ad \ ( lambda_ + -a) ^ 2 = b ^ 2 + c ^ 2 + d ^ 2 leq 1-a ^ 2 + 2ad \$$
Both conditions give the same lower limit for $$lambda _ {+}$$ (if the sign of $$a, d$$ the first inequality is equal to a lower limit and for the opposite sign, the second inequality gives a lower limit, and gives the same lower limit

So without loss of generality, I choose $$a, d geq 0$$. Then tabulating the first inequality, I get

$$lambda _ + ^ 2 – 2a lambda_ + – 1 + 2a ^ 2 + 2ad leq 0$$

This means that the maximum value of $$lambda _ + ^ {max}$$ occurs in the solution of

$$lambda _ + ^ 2 – 2a lambda_ + – 1 + 2a ^ 2 + 2ad = 0 \ Rightarrow lambda_ + = a pm sqrt {1-a ^ 2-2ad}$$

So the maximum value of $$lambda _ + ^ {max}$$ would happen in $$d = 0$$ Y $$frac {d lambda} {da} = 0 Rightarrow$$ $$lambda _ + ^ {max} = sqrt {2}$$ with $$a = frac {1} { sqrt {2}}$$

Then there is a maximum of module for all eigenvalues. Is this generally valid for a larger Hermitian matrix? If there is a defect in my referral, please let me know.