Direct Sums

There are two reasons to use the sum of two vector spaces. One of them is the way to build new vector spaces from old ones. Another reason is to decompose the known vector space into sum of two (smaller) spaces. Since we consider linear transformations between vector spaces, these sums lead to representations of these linear maps and corresponding matrices into forms that reflect these sums. In many very important situations, we start with a vector space V and can identify subspaces “internally” from which the whole space V can be built up using the construction of sums. However, the most fruitful results we obtain for a special sum, called the direct sum.

Let A and B be nonempty subsets of a vector space V. The sum of A and B, denoted A + B, is the set of all possible sums of elements from both subsets: \( A+B = \left\{ a+b \, : \, a\in A, \ b\in B \right\} . \)

A vector space V is called the direct sum of V1 and V2 if V1 and V2 are subspaces of V such that \( V_1 \cap V_2 = \{ 0 \} \) and \( V_1 + V_2 = V. \) This means that every vector v from V is uniquely represented via sum of two vectors \( {\bf v} = {\bf v}_1 + {\bf v}_2 , \quad {\bf v}_1 \in V_1 , \ {\bf v}_2 \in V_2 . \) We denote that V is the direct sum of V1 and V2 by writing \( V = V_1 \oplus V_2 . \)

Suppose that an n-dimensional vector space V is the direct sum of two subspaces \( V = U\oplus W . \) Let \( {\bf e}_1 , {\bf e}_2 , \ldots , {\bf e}_k \) be a basis of the linear subspace U and let \( {\bf e}_{k+1} , {\bf e}_{k+2} , \ldots , {\bf e}_n \) be a basis of the linear subspace W. Then \( {\bf e}_1 , {\bf e}_2 , \ldots , {\bf e}_n \) form a basis of the whole linear space V. Any linear transformation written in this basis has a matrix representation:

\[ {\bf A} = \begin{bmatrix} {\bf A}_{k \times k} & {\bf 0}_{k \times (n-k)} \\ {\bf 0}_{k \times (n-k)} & {\bf A}_{(n-k)\times (n-k)} \end{bmatrix} . \]
Therefore, the block diagonal matrix A is the direct sum of two matrices of lower sizes.
Example: Consider the Cartesian plane \( \mathbb{R}^2 , \) when every element is represented by an ordered pair v = (x,y). This vector has a unique decomposition \( {\bf v} = (x,y) = {\bf v}_1 + {\bf v}_2 = (x,0) + (0,y) , \) where vectors (x,0) and (0,y) can be identified with a one-dimensional space \( \mathbb{R}^1 = \mathbb{R} . \)

If we choose two arbitrary not parallel vectors u and v on the plane, then spans of these vectors generate two vectors spaces that we denote by U and V, respectively. Therefore, U and V are two lines containing vectores u and v, respectively. Their sum, \( U + V = \left\{ {\bf u} +{\bf v} \,: \ {\bf u} \in U, \ {\bf v} \in V \right\} \) is the whole plane \( \mathbb{R}^2 . \)

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The union \( U \cup V \) of two subspaces is not necessarily a subspace.

Example: Let E denote the set of all polynomials of even powers: \( E = \left\{ a_n t^{2n} + a_{n-1} t^{2n-2} + \cdots + a_0 \right\} , \) and O be the set of all polynomails of odd powers: \( O = \left\{ a_n t^{2n+1} + a_{n-1} t^{2n-1} + \cdots + a_0 t \right\} . \) Then the set of all polynomials P is the direct sum of these sets: \( P = O\oplus E . \)

It is easy to see that any polynomial (or function) can be ubiquely decomposed into direct sum of even and odd counterparts:

\[ p(t) = \frac{p(t) + p(-t)}{2} + \frac{p(t) - p(-t)}{2} . \]

Example: Let us consider the set M of all real (or complex) \( m \times n \) matrices, and let \( U = \left\{ {\bf A} = \left[ a_{ij} \right] :\, a_{ij} =0 \ \mbox{ for } i > j\right\} \) be the set of upper triangular matrices, and let \( W = \left\{ {\bf A} = \left[ a_{ij} \right] :\, a_{ij} =0 \ \mbox{ for } i \le j\right\} \) be the set of lower triangular matrices. Then \( M = U \oplus W . \)

Before formulating the Primary Decomposition Theorem, we need to recall some definitions and facts that were explained in other sections. We remind that the minimal polynomial of a square matrix A (or corresponding lineat transformation) is the (unique) monic polynomial ψ(λ) of least degree that annihilates the matrix A, that is ψ(A) = 0. The minimal polynomial \( \psi_u (\lambda ) \) of a vector \( {\bf u} \in V \ \mbox{ or } \ {\bf u} \in \mathbb{R}^n \) relative to A is the monic polynomial of least degree such that \( \psi_u ({\bf A}) {\bf u} = {\bf 0} . \) It follows that \( \psi_u (\lambda ) \) divides the minimal polynomial ψ(λ) of the matrix A. There exists a vector \( {\bf u} \in V (\mathbb{R}^n ) \) such that \( \psi_u (\lambda ) = \psi (\lambda ) . \) This result can be proved by representing the minimal polynomial as the product of simple terms to each of which corresponds a subspace. Then the original vector space (or \( \mathbb{R}^n \) ) is the direct sum of these subspaces.

A subspace U of a vector space V is said to be T-cyclic with respect to a linear transformation \( T\,:\, V \to V \) if there exists a vector \( {\bf u} \in U \) and a nonnegative integer r such that \( {\bf u}, T\,{\bf u} , \ldots , T^r {\bf u} \) form a basis for U. Thus, for the vector u if the degree of the minimal polynomial \( \psi_u (\lambda ) \) is k, then \( {\bf u}, T\,{\bf u} , \ldots , T^{k-1} {\bf u} \) are linearly independent and the space U is spanned by these k vectors is T-cyclic.

Theorem (Primary Decomposition Theorem): Let V be an n-dimensional vector space (n is finite) and T is a linear transformation on V. Then V is the direct sum of T-cyclic subspaces. ■

Let k be the degree of the minimal polynomial ψ(λ) of transformation T (or corresponding matrix written is specified basis), and let u be a vector in V with \( \psi_u (\lambda ) = \psi (\lambda ) . \) Then the space U spanned by \( {\bf u}, T{\bf u} , \ldots , T^{k-1} {\bf u} \) is T-cyclic. We shall prove that if \( U \ne V \quad (k \ne n), \) then there exists a T-invariant subspace W such that \( V = U\oplus W . \) Clearly, by induction on the dimension, W will then be the direct sum of T-cyclic subspaces and the proof is complete.

To show the existence of W enlarge the basis \( {\bf e}_1 = {\bf u}, {\bf e}_2 = T{\bf u} , \ldots , {\bf e}_k = T^{k-1} {\bf u} \) of U to a basis \( {\bf e}_1 , {\bf e}_2 , \ldots , {\bf e}_k , \ldots , {\bf e}_n \) of V and let \( {\bf e}_1^{\ast} , {\bf e}_2^{\ast} , \ldots , {\bf e}_k^{\ast} , \ldots , {\bf e}_n^{\ast} \) be the dual basis in the dual space. Recall that the dual space consists of all linear forms on V or, equivalently, of all functionals on V. To simplify notation, let z = ek*. Then

\[ \langle {\bf z} , {\bf e}_i \rangle =0 \quad\mbox{for } 1 \le i \le k-1 \quad\mbox{and}\quad \langle {\bf z} , {\bf e}_k \rangle =1 . \]
Consider the dual space U* spanned by \( {\bf z}, T^{\ast} {\bf z} , \ldots , T^{\ast\, k-1} {\bf z} . \) Since ψ(λ) is also the minimal polynomial of T*, the space U* is T*-invariant. Now observe that if \( U^{\ast} \cap U^{\perp} = \{ 0 \} \) and dim U* = k, then \( V^{\ast} = U^{\ast} \oplus U^{\perp} , \) where U* and \( U^{\perp} \) are T*-invariant (since dim \( U^{\perp} \) = n-k). This in turn implies the desired decomposition \( V = U^{\perp\perp} \oplus U^{\ast\perp} = U \oplus W , \) where \( U^{\perp\perp} = U \) and \( U^{\ast\perp} = W \) are T-invariant.

Finally, we shall prove that \( U^{\ast} \cap U^{\perp} = \{ 0 \} \) and dim U* = k simultaneously as follows. Suppose that \( a_0 {\bf z} + a_1 T^{\ast} {\bf z} + \cdots + a_s T^{\ast s} {\bf z} \in U^{\perp} , \) where \( a_s \ne 0 \) and \( 0 \le s \le k-1 . \) Then

\[ T^{\ast k-1-s} \left( a_0 {\bf z} + a_1 T^{\ast} {\bf z} + \cdots + a_s T^{\ast s} {\bf z} \right) = a_0 T^{\ast k-1-s} {\bf z} + a_1 T^{\ast k-s} {\bf z} + \cdots + a_s T^{\ast k-1} {\bf z} \]
is in \( U^{\perp} \) since \( U^{\perp} \) is T*-invariant. Therefore,

\[ \left\langle T^{\ast k-1-s} \left( a_0 {\bf z} + a_1 T^{\ast} {\bf z} + \cdots + a_s T^{\ast s} {\bf z} \right) , {\bf u} \right\rangle =0 . \]
This implies
\[ \left\langle {\bf z} , \left( a_0 T^{k-1-s} + a_1 T^{k-s} + \cdots + a_s T^{k-1} \right) {\bf u} \right\rangle = a_0 \left\langle {\bf z} , {\bf e}_{k-s} \right\rangle + a_1 \left\langle {\bf z} , {\bf e}_{k-s-1} \right\rangle + \cdots + a_s \left\langle {\bf z} , {\bf e}_k \right\rangle = a_s = 0 , \]
which is a contradiction. ■

Example: Consider the defective matrix (not diagonalizable)
\[ {\bf A} = \begin{bmatrix} 15&-6&2 \\ 35&-14&5 \\ 7&-3&2 \end{bmatrix} . \]
This matrix has the characteristic polynomial \( \chi (\lambda ) = \left( \lambda -1 \right)^3 \) while its minimal polynomial is \( \psi (\lambda ) = \left( \lambda -1 \right)^2 . \) The matrix A has two linearly independent eigenvectors
\[ {\bf u} = \left[ -1, 0, 7 \right]^{\mathrm T} \qquad\mbox{and} \qquad {\bf v} = \left[ 3, 7, 0 \right]^{\mathrm T} . \]
Let U and V be one-dimensional subspaces generated by spans of vectors u and v, respectively. The minimal polynomials of these vectors are the same: \( \psi_u (\lambda ) = \psi_v (\lambda ) = \lambda -1 \) because
\[ {\bf A}\,{\bf u} = {\bf u} \qquad\mbox{and} \qquad {\bf A}\,{\bf v} = {\bf v} . \]
Each of these one-dimensional subspaces U and V are A-cyclic and they cannot form the direct sum of \( \mathbb{R}^3 . \) We choose a vector \( {\bf z} = \left[ 7, -3, 1 \right]^{\mathrm T} , \) which is perpendicular to each u and v. matrix A transfers z into the vector \( {\bf A}\,{\bf z} = \left[ 125, 192, 60 \right]^{\mathrm T} , \) which is perpendicular to neither u nor v. Next application of A yields
\[ {\bf A}^2 \,{\bf z} = \begin{bmatrix} 243 \\ 587 \\ 119 \end{bmatrix} \qquad\Longrightarrow \qquad \det \begin{bmatrix} 7&125&243 \\ -3&192&587 \\ 1&60&119 \end{bmatrix} \ne 0 . \]
Hence, vectors z, Az, and A2z are linearly independent and \( \mathbb{R}^3 \) is the direct sum of A-cyclic. ■

 

Example: The infinite set of monomials \( \left\{ 1, x, x^2 , \ldots , x^n , \ldots \right\} \) form a basis in the set of all polynomials. ■