THE DIMENSION OF INTERSECTION OF k SUBSPACES

Yongge Tian

Abstract. This note presents a formula for expressing the dimension of intersection of k subspaces in an n–dimensional vector space over an arbitrary field F.

It is a well–known fundamental formula in linear algebra that for any two subspaces V_{1} and V_{2} in an m–dimensional vector space V, the dimension of the intersection V_{1} ∩ V_{2} is

dim(V_{1} ∩ V_{2}) = dim V_{1} + dim V_{2} − dim(V_{1} + V_{2}).  (1)

Written in matrix form, it is equivalent to

dim[R(A_{1}) ∩ R(A_{2})] = rk(A_{1}) + rk(A_{2}) − rk[A_{1}, A_{2}],  (2)

where A_{1} ∈ F^{m×n_{1}}, A_{2} ∈ F^{m×n_{2}}, R(·) stands for the range (column space) of a matrix.

In this note we extend it to the k–term case to establish a formula for calculating the dimension of the intersection of k column spaces R(A_{1}), R(A_{2}), …, R(A_{k}) using ranks and generalized inverses of matrices.

Let A be an m × n matrix over an arbitrary field F. A generalized inverse of A is defined as a solution of the matrix equation AXA = A and often denoted by A^{−}. For the homogeneous matrix equation AX = 0, where X ∈ F^{n×p}, its general solution can be written as X = (I_{n} − A^{−}A)U, where U ∈ F^{n×p} is arbitrary [2]. Two basic rank equalities used in the sequel related to generalized inverses [1] are

rk[A, B] = rk(A) + rk(B − AA^{−}B),  (3)

and

rk [ \begin{matrix}A \\ C \end{matrix} ] = rk(A) + rk(C − CA^{−}A).  (4)

Our main result is given below.

\underline{Theorem}. Let A_{i} ∈ F^{m×n_{i}}, i = 1, 2, …, k. Then the dimension of the intersection ∩_{i=1}^{k}R(A_{i}) is

dim(⋂_{i=1}^{k} R(A_{i})) = rk(A_{1}) + rk(A_{2}) + ⋯ + rk(A_{k}) − rk [ \begin{matrix}A_{1} & A_{2} \\ A_{1} & & A_{3} \\ ⋮ & & & ... \\ A_{1} & & & & A_{k} \end{matrix} ] .

(5)

\underline{Proof}. In order to prove (5), we first find a general expression of vectors in ∩_{i=1}^{k}R(A_{i}). Let X ∈ ∩_{i=1}^{k}R(A_{i}). Then it is obvious that there must exist X_{i} ∈ F^{n_{i}×1} such that

X = A_{1}X_{1} = A_{2}X_{2} = ⋯ = A_{k}X_{k}.  (6)

Consider it as a system of linear matrix equations, we can write it as

[ \begin{matrix}I_{m} & −A_{1} \\ I_{m} & & −A_{2} \\ ⋮ & & & ... \\ I_{m} & & & & −A_{k} \end{matrix} ] [ X ; X_{1} ; ⋮ ; X_{k} ] = 0,  (7)

or simply MY = 0. Solving for Y, we then get Y = (I_{t}−M^{−}M)U, where U ∈ F^{t×1}, t = m + n_{1} + ⋯ + n_{k}. In that case, the general expression of X is

X = [I_{m}, 0, ⋯, 0]Y = [I_{m}, 0, ⋯, 0](I_{t} − M^{−}M)U.

Thus the dimension of ∩_{i=1}^{k}R(A_{i}), by (4) is

dim(⋂_{i=1}^{k} R(A_{i})) = rk[I_{m}, 0, ⋯, 0](I_{t} − M^{−}M)

= rk [ \begin{matrix}I_{m} & 0 & ⋯ & 0 \\ I_{m} & −A_{1} \\ ⋮ & & ... \\ I_{m} & & & −A_{k} \end{matrix} ] − rk [ \begin{matrix}I_{m} & −A_{1} \\ ⋮ & & ... \\ I_{m} & & & −A_{k} \end{matrix} ]

= m + rk [ \begin{matrix}−A_{1} \\ & . & . & . \\ & & & & −A_{k} \end{matrix} ] − m − rk [ \begin{matrix}A_{1} & −A_{2} \\ A_{1} & & −A_{3} \\ ⋮ & & & ... \\ A_{1} & & & & −A_{k} \end{matrix} ]

= rk(A_{1}) + rk(A_{2}) + ⋯ + rk(A_{k}) − rk [ \begin{matrix}A_{1} & A_{2} \\ A_{1} & & A_{3} \\ ⋮ & & & ... \\ A_{1} & & & & A_{k} \end{matrix} ] ,

establishing (5).

\underline{Corollary}. Let A_{i} ∈ F^{m×n_{i}}, i = 1, 2, …, k. Then ∩_{i=1}^{k}R(A_{i}) = \{0\} holds if and only if

rk [ \begin{matrix}A_{1} & A_{2} \\ A_{1} & & A_{3} \\ ⋮ & & & ... \\ A_{1} & & & & A_{k} \end{matrix} ] = rk(A_{1}) + rk(A_{2}) + ⋯ + rk(A_{k}).  (8)

\underline{References}

1. G. Marsaglia and G. P. H. Styan, “Equalities and Inequalities for Ranks of Matrices,” Linear and Multilinear Algebra, 2 (1974), 269–292.

2. C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, New York, 1971.

Yongge Tian Department of Mathematics and Statistics Queen’s University Kingston, Ontario, Canada K7L 3N6 email: ytian@mast.queensu.ca