Identifying Cauchy Interlacing Inequalities In Hermitian Matrices

 The recent paper, Eigenvalues from Eigenvectors: A study of Basic Identity in Linear Algebra  (Bull. Am. Math. Society, Vol. 59, No. 1, January 2022, p. 31) presents an excellent launch point to investigate aspects of linear algebra, especially the role of Cauchy interlacing inequalities in obtaining eigenvalues from eigenvectors in Hermitian matrices.  

Basically, a square matrix (one with the same number of rows and columns) is called Hermitian if it has complex numbers, and if:

A = A*

i.e. the matrix is found equal to its conjugate transpose. Basically, a square matrix (one with the same number of rows and columns) is called Hermitian if it has complex numbers, and if:  A = A*

Example: Show that the matrix M =

(2.....i)
(-i....5)

is Hermitian

Solution:

We find the complex conjugate of the matrix or, M' =

(2.....-i)
(i......5)

Then we obtain the transpose of that, M* = t^(M') =

(2.....i)
(-i......5)

and since M = M* = t^(M')

the matrix is Hermitian.

If  A is an n x n Hermitian matrix, denote its n real eigenvalues by

l₁ (A)….. l_n (A)   

And for the sake of convenience we adopt the convention of non-decreasing eigenvalues such that:

l₁ (A)  <….. < l_n (A)  

 

Further, if 1 <  j <  n  let   M _j  denote the (n- 1) by (n -1) minor formed from A by deleting the jth row and column from A. This is again a Hermitian matrix and has n – 1 real eigenvalues

l ₁ (M _j),……. l_n-1 (M _j)

Which for sake of concreteness we arrange in non-decreasing order.  Then we will have the Cauchy interlacing inequalities:

l _i (A _j) <  l _i (M _j)  <   l _i+1 (A)

For i= 1,…….n – 1

The Cauchy interlace theorem, then, states basically that the eigenvalues of a Hermitian matrix A of order n are interlaced with those of any principal submatrix of order n − 1.

We will let P _A (l) be the characteristic polynomial from which the eigenvectors and eigenvalues are obtained.  Following the paper :

Let A =

(1.....1......-1)
(1.....3... ...1)
(-1.....1......3)

From which we obtain:

(1 - l…..1…….-1)

(0……. 3 - l ....  1)

(-1…….. 1… 3- l )

Then the minors and corresponding eigenvalues  are given by:

M ₁  = 

(3...... 1)

(1.......3)

l_1,2 (M₁)  =  2, 4

M ₂  = 

(1.....-1)

(-1....3)

 l_1,2 (M ₂)  =  2 +  Ö2  » 0.39,   3.4

M ₃  = 

(1.....1)

(1.....3)

l_1,2 (M ₂)  =  2 +  Ö2  » 0.39,   3.4  

Then the eigenvectors are:

With corresponding eigenvalues:

l₁ (A) =  0,    l ₂ (A) =  3,   l₃ (A)  =  4

From which one can observe the interlacing inequalities.

Suggested Problems:  

1) Determine whether the matrix Y =

(1.....(1+ 1i).......5)
((1- i).......2... ...i)
(5..........-i..........7)

Is Hermitian

2) From the mathematical development in the post, indicate two examples where the Cauchy interlace theorem applies.  I.e. give the specific submatrices of order n − 1 related to the defined Hermitian matrix of order n, and confirm the eigenvalues are real.

3)Let P _A (l) =  

(l- a ₁.....0.........0)

(0.......l – a ₂.....0)

(0.......0......l - a _n)

Show how one can set out the characteristic equation and thereby obtain the eigenvalues by solving for a₁, a ₂, a _n,