This page contains corrections and comments for the book Matrix Groups: An Introduction to Lie Group Theory published by SpringerVerlag. Please send me any observations, corrections, etc. using the following form and I will endeavour to publish a suitable response.
Page 68, line 3: the Jacobi identity should be
[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0.
Page 106, Prop. 4.21: u statement should end with `...and each Ae_i is a simple algebra.'.
Page 140, line 1: u should be an element of Gamma_n.
Line 4: `Chapters 1,3,7' should be `Chapters 1,2,3,7'.
Second `M_n(R)' should be `M_n(C)'.
`domain' should be `codomain'.
the first term should be `$rho(x,s_n)'.
The end of the equation should be `...=det^{1}R^+'.
The end of the sequence of equations should be `...=exp(A+B)'.
This result would probably have been better put later in the chapter after the material on oneparameter subgroups. Here is a rewritten version (07/05/2002) with some more details about what goes into the proof which follows that outlined by G. Segal in [4], page 80.
The expression at the end of the line should be `J(lambda_j,r_j)'.
The equation should be `gamma'(t)=gamma(t)A'.
\paragraph{p}The intended value of v(0) was [0 0 1]^T, and in the Solution the (2,2) term of the matrix exp(tS) needs an extra factor of 5. Here is a rewritten version (14/05/2002).
The last equation should read `...=ce^{2t}'.
The second `={0}' should be deleted.
Equation should read `...with alpha(0)=I'.
It should read `dim SkHerm_n(C)=n+2....'.
`ker Ad' should be just `Ad'.
It should be `U_2 = x_2 H^ + y_2 E^ + z_2 F^'.
Final `H^' should be `H'.
`cos' should be `cos phi'.
The left hand side of the equation should be `av_j'.
This should read `If a in ker lambda then a1=0, so a=0.'
This should read `(1,1) in ker lambda^times times rho^times, 1 in ker Delta.
In the expression for $e_3$, 1/6 should be 1/3.
`nu(a,b)<=nu(a)nu(b)' should be `nu(a,b)=nu(a)nu(b)''.
The sum on the left side should be from 1 to m, not 1 to n, and the sum of the right side should be over r not s. In the next equation, the x's should go from 1 to m.
`... k = i = kj ...' should be `... jk = i = kj ...'.
`z=' should be `z=u1 + vi'.
The middle equation should read `alpha(j)=Aj'.
The equation should read `...=dim Symp_{2n}(R)'.
`e_se_r = e_se_r' should be `e_se_r = e_re_s'.
The second `=' is missing.
`U_k=...' should be `U_j=...'.
In the equation for <U_j,U_k>, the last case should read `if j=k=n+1'.
The 2 times 2 matrix should be R(theta) (with no subscript).
In the definition of U'', the factor in front should be `1/2i'.
Should read `the Lie algebra of'.
This should read `A topological space X is separable if it has a countable basis, i.e., a basis of the form...'.
Should read `...a collection of ncharts'.
The domain of `f_0 circ gamma' should be`(a,b) cap gamma^{1}f_0^{1}V_0'.
The domain of `f_0 circ gamma' should be `(a,b) cap gamma^{1}f^{1}V_0'.
`(f\gamma)'(0)' should be `(f_0\gamma)'(0)'.
In each case, the sums should be over `k=>2' rather than `k<=2'.
1/2C^2 should be 1/2C.
In each case the second vector should be `(0,1,0)'.
Should read `...which shows that phi(g) v is another'.
Should read `...there are nonzero vectors'.
`pi U subset G' should be `pi U subset G/H'.
The right hand arrow should be labelled with `f' and the bottom arrow with `overline{f}'.
`p(0)=x and q(0)=y' should be `p(1)=x
and q(1)=y'.
Page 308, line 1:
The (2,1) entry `2n' in the matrix on the right hand side should be `2nt^{n1}'.
Yes I know, Principle in Theorems 10.13 & 10.15 (and anywhere else) should be Principal.
Thanks to R. Barraclough, R. Chapman, P. Eccles, Robin Endelman, S. Hendren, N. Pollock, M. Yankelevitch and R. Wickner for spotting the above.
