Say A=B*B^H (^H means conjugate transition)
C=A - (diagonal of A).
So C is a symmetric matrix which diagonal entries are all 0.
I wondering whether there is any study on how to judge C is positive definite, or not
Thanks,
Happy Thanksgiving!
Evaluate the eigenvalues of C. Are they all positive? (With a zero diagonal I am thinking there is no way that C is positive definite).
Thank you, Prof. Morton E. O'Kelly.
Eigenvalues can help me evaluate C. I am wonder whether there is a general criteria by which the C is positive definite or negative definite? or any on-going discussion on this? Thanks,
Regards,
Pengda
I don't have an answer for a generalized case. However, for 2 x 2 matrices and 3 x 3 matrices with real entries , they are neither positive definite nor negative definite.
For example,
If you consider C = [0 a;a 0], then Eigenvalues of C is +a or -a, so neither positive definite nor negative definite.
If you consider C = [0 a b; a 0 c; b c 0], then the characteristic equation to find the Eigenvalues is -l^3+(a^2+b^2+c^2)l+abc=0. From the relation between roots and coefficients, it is clear that l1+l2+l3=0, l1l2+ l1l3+ l3l2=-(a2+b2+c2) and l1l2 l3=abc. From the second relation, it is clear that, it is neither positive nor negative.
Sum of eigenvalues is equal to trace of the matrix which is zero. Therefore, there is at least one eigenvalue which is negative.Hence the matrix is not positive definite.
If a real or complex matrix is positive definite, then all of its principal minors are positive. Since the diagonal entries are the also the one-by-one principal minors of a matrix, any matrix with a diagonal entry equal to zero cannot be positive definite. (And it cannot be negative definite, either.) At best, the matrix will be positive semidefinite. Consider the matrix [ 5 0; 0 0], with eigenvalues 5 and 0. Clearly, this matrix is positive semidefinite, but not positive definite.
Dear Jeffery: Considered matrix has a non-zero value in the diagonal. For a two by two real matrix with zero diagonals, it will be positive semidefinite, iff the matrix is a zero matrix.
Panchatcharam
You may have misunderstood my point: EVEN A SINGLE ZERO ON THE DIAGONAL prevents a real, symmetric matrix from being positive definite. I would agree with you that the only 2x2 symmetric, positive semidefinite matrix with all diagonal entries zero is the 2x2 zero matrix.
In fact, ANY n x n real, symmetric matrix with ALL diagonal entries zero is positive semidefinite ONLY when the matrix is the n x n zero matrix. This follows from the fact that every 2x2 principal submatrix of a positive semidefinite matrix is itself a positive semidefinite matrix. From the previous observation, each 2x2 principal submatrix must be the 2x2 zero matrix, and hence, every entry of the nxn matrix is zero.
By the way, the result for negative semidefiniteness is not the same. The 2 x 2 matrix [0 1; 1 0] is real, symmetric, has all diagonal entries zero, AND is negative semidefinite since the 1 x 1 principal minors are zero, and the 2x2 principal minor (ie. the determinant of the matrix) is negative.
I hope that this clears up the matter for you.
Dear Jeffery: Thanks for the clarification and detailed answer.
Theorem (Sylvester's Criterion). A symmetric matrix A in Mn(R) is positive definite if and only if its leading principal minors are positive, that is, det(Ak) > 0, for all k.
Therefore a symmetric matrix with zero diagonal entries is not positive definite, since det(A1) = 0.
I have a matrix which has a positive definiteness of 0, and it has both negative and positive complex conjugates in the eigenvalues. What would this matrix be classified as in terms of definiteness?
Sergio
How are you defining positive definiteness? Real? Symmetric? Or solely. x^* M x > 0 for all column vectors x In C^n. ? What do you mean by a “positive definiteness of 0”?
if Mx = £x for a nonzero real or complex vector x and a real or complex number £, then x^* x > 0. consequently, x^* Mx = x^* £x = £ x^* x is a positive multiple of £. If M is positive definite (by essentially all definitions of positive definiteness), this forces £ to be positive. That is a positive definite matrix must have all eigenvalues real AND POSITIVE.
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