Implications of Symmetry

A real n by n matrix A is defined to be symmetric if, for every possible pair of indices i and j between 1 and n, it is the case that A_i,j = A_j,i.

In order to give you some practice with this concept, and with manipulating matrices as abstract quantities, we offer a list of tasks. In each case, we state a fact about a matrix A or a formula for its entries, and it is your problem to demonstrate that this information implies that A must be symmetric.

The notation I indicates the identity matrix; x' is the transpose of the vector x; similarly, A' is the transpose of the matrix A. We assume all constants, vectors and matrices in the following discussion have real entries.

1. A_i,j = i * ( n - j + 1 ) / ( n + 1 ) if i <= j, or else j * ( n - i + 1 ) / ( n + 1 ) if j < i.
2. A = c₁ * x₁ * x'₁ + c₂ * x₂ * x'₂ + ... + c_n * x₁ * x'_n for some real constants c_i and n-vectors x_i.
3. (u,v) = u' * (A*v) defines an inner product for all n-vectors u and v.
4. A = Q * D * inverse(Q) where D is a diagonal matrix and Q is an orthogonal matrix.
5. A = I - 2 * x * x' for some n-vector x.
6. The inverse of A exists, and is a symmetric matrix.
7. A_i,j = 1 if i+j=n+1, 0 otherwise.
8. A_i,j = | x(i) - x(j) | for some given n-vector x.
9. A_i,j = 2 * min ( i, j ) - 1.
10. A_i,j = alpha^|i-j| for some real number alpha.
11. A_i,j = min ( i, j ) / max ( i, j ).
12. A_1,1 = 1,
    A_i,j = A_i-1,j + A_i,j-1
    with "out-of-range" entries taken to be 0.
13. The matrix exponential e^(At) is a symmetric matrix.
14. A * x = A' * x for every n-vector x.
15. A is the Hessian matrix of a real-valued function f(x), where x is a real n-vector argument, so that A_i,j = d^2 f(x) dxi dxj.
16. The singular value decomposition of a matrix A is defined to have the form A = U * S * V', where U and V are orthogonal, and S is diagonal. For our particular matrix A, the SVD happens to have the property that U = V.
17. Every nonzero vector x has a strictly positive projection onto its image A*x.

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