Friday, February 3, 2023
HomeMatlabWhat Is the Hint of a Matrix? – Nick Higham

What Is the Hint of a Matrix? – Nick Higham


The hint of an ntimes n matrix is the sum of its diagonal components: mathrm{trace}(A) = sum_{i=1}^n a_{ii}. The hint is linear, that’s, mathrm{trace}(A+B) = mathrm{trace}(A) + mathrm{trace}(B), and mathrm{trace}(A) = mathrm{trace}(A^T).

A key truth is that the hint can be the sum of the eigenvalues. The proof is by contemplating the attribute polynomial p(t) = det(t I - A) = t^n + a_{n-1}t^{n-1} + dots + a_1 t + a_0. The roots of p are the eigenvalues lambda_1, lambda_2, dots, lambda_n of A, so p will be factorized

notag  p(t) = (t - lambda_1) (t - lambda_2) dots (t - lambda_n),

and so a_{n-1} = -(lambda_{11} + lambda_{22} + cdots + lambda_{nn}). The Laplace enlargement of det(t I - A) reveals that the coefficient of t^{n-1} is -(a_{11} + a_{22} + cdots + a_{nn}). Equating these two expressions for a_{n-1} offers

notag     mathrm{trace}(A) = displaystylesum_{i=1}^n lambda_i.  qquadqquad(1)

A consequence of (1) is that any transformation that preserves the eigenvalues preserves the hint. Due to this fact the hint is unchanged underneath similarity transformations: mathrm{trace}(X^{-1}AX) = mathrm{trace}(A) for any nonsingular X.

An an instance of how the hint will be helpful, suppose A is a symmetric and orthogonal ntimes n matrix, in order that its eigenvalues are pm 1. If there are p eigenvalues 1 and q eigenvalues -1 then mathrm{trace}(A) = p - q and n = p + q. Due to this fact p = (n + mathrm{trace}(A))/2 and q = (n - mathrm{trace}(A))/2.

One other necessary property is that for an mtimes n matrix A and an ntimes m matrix B,

notag  mathrm{trace}(AB) = mathrm{trace}(BA) qquadqquad(2)

(although AB ne BA basically). The proof is easy:

notag begin{aligned}   mathrm{trace}(AB) &= displaystylesum_{i=1}^m (AB)_{ii}                      = sum_{i=1}^m sum_{k=1}^n a_{ik} b_{ki}                      = sum_{k=1}^n sum_{i=1}^m b_{ki} a_{ik}                       & = sum_{k=1}^n (BA)_{kk}                      = mathrm{trace}(BA). end{aligned}

This straightforward truth can have non-obvious penalties. For instance, take into account the equation AX - XA = I in ntimes n matrices. Taking the hint offers 0 = mathrm{trace}(AX) - mathrm{trace}(XA)    = mathrm{trace}(AX - XA)    = mathrm{trace}(I) = n, which is a contradiction. Due to this fact the equation has no resolution.

The relation (2) offers mathrm{trace}(ABC) = mathrm{trace}((AB)C) = mathrm{trace}(C(AB)) = mathrm{trace}(CAB) for ntimes n matrices A, B, and C, that’s,

notag   mathrm{trace}(ABC) = mathrm{trace}(CAB). qquadqquad(3)

So we will cyclically permute phrases in a matrix product with out altering the hint.

For instance of the usage of (2) and (3), if x and y are n-vectors then mathrm{trace}(xy^T) = mathrm{trace}(y^Tx) = y^Tx. If A is an ntimes n matrix then mathrm{trace}(xy^TA) will be evaluated with out forming the matrix xy^TA since, by (3), mathrm{trace}(xy^TA) =                 mathrm{trace}(y^TAx) = y^TAx.

The hint is helpful in calculations with the Frobenius norm of an mtimes n matrix:

notag     |A|_F = left(displaystylesum_{i=1}^m sum_{i=1}^n |a_{ij}|^2right)^{1/2}             = bigr(mathrm{trace}(A^*A)bigr)^{1/2},

the place * denotes the conjugate transpose. For instance, we will generalize the formulation |x+mathrm{i}y|^2 = x^2 + y^2 for a fancy quantity to an mtimes n matrix A by splitting A into its Hermitian and skew-Hermitian elements:

notag  A = frac{1}{2}(A+A^*) + frac{1}{2}(A-A^*) equiv B + C,

the place B = B^* and C = -C^*. Then

notag begin{aligned}     |A|_F^2 =     |B + C|_F^2 &= mathrm{trace}((B+C)^*(B+C))                 &= mathrm{trace}(B^*B + C^*C) + mathrm{trace}(B^*C + C^*B)                 &= mathrm{trace}(B^*B + C^*C) + mathrm{trace}(BC - CB)                 &= mathrm{trace}(B^*B + C^*C)                 &= |B|_F^2 + |C|_F^2. end{aligned}

If a matrix is just not explicitly identified however we will compute matrix–vector merchandise with it then the hint will be estimated by

notag      mathrm{trace}(A) approx x^TAx,

the place the vector x has components independently drawn from the usual regular distribution with imply 0 and variance 1. The expectation of this estimate is

notag begin{aligned}     Ebigl( x^TAx bigr)  &=     Ebigl( mathrm{trace}(x^TAx) bigr)  =     Ebigl( mathrm{trace}(Axx^T) bigr)  =     mathrm{trace} bigl( E(Axx^T) bigr)       &=  mathrm{trace}bigl(A E(xx^T) bigr)       = mathrm{trace}(A), end{aligned}

since E(x_ix_j) = 0 for ine j and E(x_i^2) = 1 for all i, so E(xx^T) = I. This stochastic estimate, which is because of Hutchinson, is subsequently unbiased.

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