There is now a certain amount of routine work to be done, most of which we shall leave to the imagination. The problem is this: in a fixed coordinate system \(\mathcal{X}=\{x_{1}, \ldots, x_{n}\}\) , knowing the matrices of \(A\) and \(B\) , how can we find the matrices of \(\alpha A+\beta B\) , of \(A B\) , of \(0\) , \(1\) , etc.?

Write \([A]=(\alpha_{i j})\) , \([B]=(\beta_{i j})\) , \(C=\alpha A+\beta B\) , \([C]=(\gamma_{i j})\) ; we assert that \[\gamma_{i j}=\alpha \alpha_{i j}+\beta \beta_{i j};\] also if \([0]=(o_{i j})\) and \([1]=(e_{i j})\) , then \[o_{i j}=0\] and

\[e_{i j}=\delta_{i j} \text {(= the Kronecker delta).}\] 

A more complicated rule is the following: if \(C=A B\) , \([C]=(\gamma_{i j})\) , then \[\gamma_{i j}=\sum_{k} \alpha_{i k} \beta_{k j}.\] To prove this we use the definition of the matrix associated with a transformation, and juggle, thus: \begin{align} C x_{j} = A(B x_{j}) &= A\Big(\sum_{k} \beta_{k j} x_{k}\Big)\\ &= \sum_{k} \beta_{k j} A x_{k}\\ &= \sum_{k} \beta_{k j}\Big(\sum_{i} \alpha_{i k} x_{i}\Big)\\ &=\sum_{i}\Big(\sum_{k} \alpha_{i k} \beta_{k j}\Big) x_{i}. \end{align} 

The relation between transformations and matrices is exactly the same as the relation between vectors and their coordinates, and the analogue of the isomorphism theorem of Section: Isomorphism is true in the best possible sense. We shall make these statements precise.

With the aid of a fixed basis \(\mathcal{X}\) , we have made correspond a matrix \([A]\) to every linear transformation \(A\) ; the correspondence is described by the relations \(A x_{j}=\sum_{i} \alpha_{i j} x_{i}\) . We assert now that this correspondence is one-to-one (that is, that the matrices of two different transformations are different), and that every array \((\alpha_{i j})\) of \(n^{2}\) scalars is the matrix of some transformation. To prove this, we observe in the first place that knowledge of the matrix of \(A\) completely determines \(A\) (that is, that \(A x\) is thereby uniquely defined for every \(x\) ), as follows: if \(x=\sum_{j} \xi_{j} x_{j}\) , then \begin{align} A x &= \sum_{j} \xi_{j} A x_{j}\\ &= \sum_{j} \xi_{j}\Big(\sum_{i} \alpha_{i j} x_{i}\Big)\\ &= \sum_{i}\Big(\sum_{j} \alpha_{i j} \xi_{j}\Big) x_{i}. \end{align} (In other words, if \(y=A x=\sum_{i} \eta_{i} x_{i}\) , then \(\eta_{i}=\sum_{j} \alpha_{i j} \xi_{j}.\) Compare this with the comments in Section: Matrices on the perversity of indices.) In the second place, there is no law against reading the relation \(A x_{j}=\sum_{i} \alpha_{i j} x_{i}\) backwards. If, in other words, \((\alpha_{i j})\) is any array, we may use this relation to define a linear transformation \(A\) ; it is clear that the matrix of \(A\) will be exactly \((\alpha_{i j})\) . (Once more, however, we emphasize the fundamental fact that this one-to-one correspondence between transformations and matrices was set up by means of a particular coordinate system, and that, as we pass from one coordinate system to another, the same linear transformation may correspond to several matrices, and one matrix may be the correspondent of many linear transformations.) The following statement sums up the essential part of the preceding discussion.

Theorem 1. Among the set of all matrices \((\alpha_{i j})\) , \((\beta_{i j})\) , etc., \(i, j=1, \ldots, n\) (not considered in relation to linear transformations), we define sum, scalar multiplication, product, \((o_{i j})\) , and \((e_{i j})\) , by \begin{align} (\alpha_{i j})+(\beta_{i j}) & =(\alpha_{i j}+\beta_{i j}) \\ \alpha(\alpha_{i j}) & =(\alpha \alpha_{i j}) \\ (\alpha_{i j})(\beta_{i j}) & =\Big(\sum_{k} \alpha_{i k} \beta_{k j}\Big) \\ o_{i j} & =0,\\ e_{i j} & =\delta_{i j} . \end{align} Then the correspondence (established by means of an arbitrary coordinate system \(\mathcal{X} = \{x_{1}, \ldots, x_{n}\}\) of the \(n\) -dimensional vector space \(\mathcal{V}\) ), between all linear transformations \(A\) on \(\mathcal{V}\) and all matrices \((\alpha_{i j})\) , described by \(A x_{j}=\sum_{i} \alpha_{i j} x_{i}\) , is an isomorphism; in other words, it is a one-to-one correspondence that preserves sum, scalar multiplication, product, \(0\) , and \(1\) .

We have carefully avoided discussing the matrix of \(A^{-1}\) . It is possible to give an expression for \([A^{-1}]\) in terms of the elements \(\alpha_{i j}\) of \([A]\) , but the expression is not simple and, fortunately, not useful for us.

EXERCISES

Exercise 1. Let \(A\) be the linear transformation on \(\mathcal{P}_{n}\) defined by \((A x)(t)=x(t+1)\) , and let \(\{x_{0}, \ldots, x_{n-1}\}\) be the basis of \(\mathcal{P}_{n}\) defined by \(x_{j}(t)=t^{j}\) , \(j=0, \ldots, n-1\) . Find the matrix of \(A\) with respect to this basis.

Exercise 2. Find the matrix of the operation of conjugation on \(\mathbb{C}\) , considered as a real vector space, with respect to the basis \(\{1, i\}\) (where \(i=\sqrt{-1}\) ).

Exercise 3. 

  1. Let \(\pi\) be a permutation of the integers \(1, \ldots, n\) ; if \(x=(\xi_{1}, \ldots, \xi_{n})\) is a vector in \(\mathbb{C}^{n}\) , write \(A x=(\xi_{\pi(1)}, \ldots, \xi_{\pi(n)})\) . If \(x_{i}=(\delta_{i 1}, \ldots, \delta_{i n})\) , find the matrix of \(A\) with respect to \(\{x_{1}, \ldots, x_{n}\}\) .
  2. Find all matrices that commute with the matrix of \(A\) .

Exercise 4. Consider the vector space consisting of all real two-by-two matrices and let \(A\) be the linear transformation on this space that sends each matrix \(X\) onto \(P X\) , where \[P=\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}.\] Find the matrix of \(A\) with respect to the basis consisting of \[\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 1\\ 0 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0\\ 1 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}.\] 

Exercise 5. Consider the vector space consisting of all linear transformations on a vector space \(\mathcal{V}\) , and let \(A\) be the (left) multiplication transformation that sends each transformation \(X\) on \(\mathcal{V}\) onto \(P X\) , where \(P\) is some prescribed transformation on \(\mathcal{V}\) . Under what conditions on \(P\) is \(A\) invertible?

Exercise 6. Prove that if \(I\) , \(J\) , and \(K\) are the complex matrices \[\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}, \quad \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}, \quad \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix}\] respectively (where \(i=\sqrt{-1}\) ), then \begin{align} I^{2} &= J^{2}=K^{2}=-1,\\ I J &=-J I=K,\\ J K &=-K J=I,\\ K I &=-I K=J. \end{align} 

Exercise 7. 

  1. Prove that if \(A\) , \(B\) , and \(C\) are linear transformations on a two-dimensional vector space, then \((A B-B A)^{2}\) commutes with \(C\) .
  2. Is the conclusion of (a) true for higher-dimensional spaces?

Exercise 8. Let \(A\) be the linear transformation on \(\mathbb{C}^{2}\) defined by \(A(\xi_{1}, \xi_{2})=(\xi_{1}+\xi_{2}, \xi_{2})\) . Prove that if a linear transformation \(B\) commutes with \(A\) , then there exists a polynomial \(p\) such that \(B=p(A)\) .

Exercise 9. For which of the following polynomials \(p\) and matrices \(A\) is it true that \(p(A)=0\) ?

  1. \(p(t)=t^{3}-3 t^{2}+3 t-1\) , \(A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}\) .
  2. \(p(t)=t^{2}-3 t\) , \(A= \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}\) .
  3. \(p(t)=t^{3}+t^{2}+t+1\) , \(A=\begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{bmatrix}\) .
  4. \(p(t)=t^{3}-2 t\) , \(A=\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}\) .

Exercise 10. Prove that if \(A\) and \(B\) are the complex matrices \[\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{bmatrix} \quad \text { and } \quad \begin{bmatrix} i & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -i & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\] respectively (where \(i=\sqrt{-1}\) ), and if \(C=A B-i B A\) , then \(C^{2}+C^{2}+C=0\) .

Exercise 11. If \(A\) and \(B\) are linear transformations on a vector space, and if \(A B=0\) , does it follow that \(B A=0\) ?

Exercise 12. What happens to the matrix of a linear transformation on a finite-dimensional vector space when the elements of the basis with respect to which the matrix is computed are permuted among themselves?

Exercise 13. 

  1. Suppose that \(\mathcal{V}\) is a finite-dimensional vector space with basis \(\{x_{1}, \ldots, x_{n}\}\) . Suppose that \(\alpha_{1}, \ldots, \alpha_{n}\) are pairwise distinct scalars. If \(A\) is a linear transformation such that \(A x_{j}=\alpha_{j} x_{j}\) , \(j=1, \ldots, n\) , and if \(B\) is a linear transformation that commutes with \(A\) , then there exist scalars \(\beta_{1}, \ldots, \beta_{n}\) such that \(B x_{j}=\beta_{j} x_{j}\) .
  2. Prove that if \(B\) is a linear transformation on a finite-dimensional vector space \(\mathcal{V}\) and if \(B\) commutes with every linear transformation on \(\mathcal{V}\) , then \(B\) is a scalar (that is, there exists a scalar \(\beta\) such that \(B x=\beta x\) for all \(x\) in \(\mathcal{V}\) ).

Exercise 14. If \(\{x_{1}, \ldots, x_{k}\}\) and \(\{y_{1}, \ldots, y_{k}\}\) are linearly independent sets of vectors in a finite-dimensional vector space \(\mathcal{V}\) , then there exists an invertible linear transformation \(A\) on \(\mathcal{V}\) such that \(A x_{j}=y_{j}\) , \(j=1, \ldots, k\) .

Exercise 15. If a matrix \([A]=(\alpha_{i j})\) is such that \(\alpha_{i i}=0\) , \(i=1, \ldots, n\) , then there exist matrices \([B]=(\beta_{i j})\) and \([C]=(\gamma_{i j})\) such that \([A]=[B][C]-[C][B]\) . (Hint: try \(\beta_{i j}=\beta_{i} \delta_{i j}\) .)

Exercise 16. Decide which of the following matrices are invertible and find the inverses of the ones that are.

2 

  1. \(\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\) .
  2. \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) .
  3. \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}\) .
  4. \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) .
  5. \(\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}\) .
  6. \(\begin{bmatrix} 1 & 0 & 1 \\ 1 & 0 & 1 \\ 1 & 0 & 1 \end{bmatrix}\) .
  7. \(\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}\) .

Exercise 17. For which values of \(\alpha\) are the following matrices invertible? Find the inverses whenever possible.

2 

  1. \(\begin{bmatrix} \alpha & 1 \\ 1 & 0 \end{bmatrix}\) .
  2. \(\begin{bmatrix} 1 & \alpha \\ 1 & 0 \end{bmatrix}\) .
  3. \(\begin{bmatrix} 1 & \alpha \\ 1 & \alpha \end{bmatrix}\) .
  4. \(\begin{bmatrix} 1 & 1 \\ 1 & \alpha \end{bmatrix}\) .

Exercise 18. For which values of \(\alpha\) are the following matrices invertible? Find the inverses whenever possible.

2 

  1. \(\begin{bmatrix} 1 & \alpha & 0 \\ \alpha & 1 & \alpha \\ 0 & \alpha & 1 \end{bmatrix}\) .
  2. \(\begin{bmatrix} \alpha & 1 & 0 \\ 1 & \alpha & 1 \\ 0 & 1 & \alpha \end{bmatrix}\) .
  3. \(\begin{bmatrix} 0 & 1 & \alpha \\ 1 & \alpha & 0 \\ \alpha & 0 & 1 \end{bmatrix}\) .
  4. \(\begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & \alpha \\ 1 & \alpha & 1 \end{bmatrix}\) .

Exercise 19. 

  1. It is easy to extend matrix theory to linear transformations between different vector spaces. Suppose that \(\mathcal{U}\) and \(\mathcal{V}\) are vector spaces over the same field, let \(\{x_{1}, \ldots, x_{n}\}\) and \(\{y_{1}, \ldots, y_{m}\}\) be bases of \(\mathcal{U}\) and \(\mathcal{V}\) respectively, and let \(A\) be a linear transformation from \(\mathcal{U}\) to \(\mathcal{V}\) . The matrix of \(A\) is, by definition, the rectangular, \(m\) by \(n\) , array of scalars defined by \[A x_{j}=\sum_{i} \alpha_{i j} y_{i}.\] Define addition and multiplication of rectangular matrices so as to generalize as many as possible of the results of Section: Matrices of transformations . (Note that the product of an \(m_{1}\) by \(n_{1}\) matrix and an \(m_{2}\) by \(n_{2}\) matrix, in that order, will be defined only if \(n_{1}=m_{2}\) .)
  2. Suppose that \(A\) and \(B\) are multipliable matrices. Partition \(A\) into four rectangular blocks (top left, top right, bottom left, bottom right) and then partition \(B\) similarly so that the number of columns in the top left part of \(A\) is the same as the number of rows in the top left part of \(B\) . If, in an obvious shorthand, these partitions are indicated by \[A=\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, \quad B=\begin{bmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{bmatrix},\] then \[A B=\begin{bmatrix} A_{11} B_{11}+A_{12} B_{21} & A_{11} B_{12}+A_{12} B_{22} \\ A_{21} B_{11}+A_{22} B_{21} & A_{21} B_{12}+A_{22} B_{22} \end{bmatrix}\] 
  3. Use subspaces and complements to express the result of (b) in terms of linear transformations (instead of matrices).
  4. Generalize both (b) and (c) to larger numbers of pieces (instead of four).