In Axler's "Linear Algebra Done Right", he gave the singular-value decomposition as:
$Tv = s_1\langle v,e_1\rangle f_1 + \cdots + s_n\langle v,e_n\rangle f_n$, where T is an operator; $s_1,\ldots,s_n$ are the singular values of $T$; $(e_1,\ldots,e_n)$ and $(f_1,\ldots,f_n)$ are orthonormal bases of $V$. Singular values of $T$ are defined as eigenvalues of $\sqrt{T^*T}$. $\langle ,\rangle $ is inner product.
I am having trouble connecting this to the more generally used matrix representation $A=UΣV^T$. Matrices in Axler's book are considered as linear maps, "with respect to some bases". How do I understand Axler's exposition using matrices?
Thank you.
Update: To clarify, I was thinking from the perspective where an operator is a matrix with respect to some basis (in his book...). So an operator can be written as $M(T,(b1,...bn),(b1,...bn))$. If I apply change of basis, I get $M(I,(f1,...fn),(b1,...bn))*M(T,(e1,...en),(f1,...fn))*M(I,b1,...bn),(e1,...en))$ where (e1,...en) is an orthonormal basis; (f1,...fn) is another orthonormal basis given by $fi=Sei$ given by isometry $S$ from polar decomposition.
$M(T,(e1,...en),(f1,...fn))$ gives me the diagonol matrix $Σ$. But how about the rest? Thanks.
Update 2: Thanks for the answers below and I understand those. But my question is a bit specific to Axler's definition. In Axler's book, he thinks of matrix a bit differently: m*n matrix is defined as $M(T,(v_1...v_n),(w_1..w_m))$ where $T$ is a linear map from space $V$ to W, and $(v_1...v_n)$ and $(w_1...w_m)$ are bases for 2 spaces. For the $k$th column in the matrix, each entry $a_{ik}$ is defined as $Tv_k=a_{1k}w_1+...a_{mk}w_m$, so the matrix represents the linear map between the bases. So it's like "you don't talk about a matrix without talking about $T$ and the bases". Whether this is a good way to think about matrix is probably also a topic for debate...
Anyway, now with that, I am trying to decompose: $M(T,(b_1,...b_n),(b_1,...b_n))$. I already know that the diagonol $Σ$ in SVD can be thought of as: A. $M(T,(e_1,...e_n),(f_1,...f_n))$, (where $f_i=Se_i$ given by isometry $S$ from polar decomposition $T=S\sqrt{T^*T}$), OR, B. $M(\sqrt{T^*T},(e_1,...e_n),(e_1,...e_n))$. So I would also like to know: 1. the operator 2. the bases, for the other 2 unitary matrices, no matter which one you choose for $Σ$.
Thanks again.