In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
In my math lectures, we talked about the Gram-Determinant where a matrix times its transpose are multiplied together.
Is $A A^\mathrm T$ something special for any matrix $A$?
The main thing is presumably that $AA^T$ is symmetric. Indeed $(AA^T)^T=(A^T)^TA^T=AA^T$. For symmetric matrices one has the Spectral Theorem which says that we have a basis of eigenvectors and every eigenvalue is real.
Moreover if $A$ is invertible, then $AA^T$ is also positive definite, since $$x^TAA^Tx=(A^Tx)^T(A^Tx)> 0$$
Then we have: A matrix is positive definite if and only if it's the Gram matrix of a linear independent set of vectors.
Last but not least if one is interested in how much the linear map represented by $A$ changes the norm of a vector one can compute
$$\sqrt{\left<Ax,Ax\right>}=\sqrt{\left<A^TAx,x\right>}$$
which simplifies for eigenvectors $x$ to the eigenvalue $\lambda$ to
$$\sqrt{\left<Ax,Ax\right>}=\sqrt \lambda\sqrt{\left<x,x\right>},$$
The determinant is just the product of these eigenvalues.
$AA^T$ is positive semi-definite, and in a case in which $A$ is a column matrix, it will be a rank 1 matrix and have only one non-zero eigenvalue which equal to $A^TA$ and its corresponding eigenvector is $A$. The rest of the eigenvectors are the null space of $A$ i.e. $\lambda^TA = 0$.
Indeed, independent of the size of $A$, there is a useful relation in the eigenvectors of $AA^T$ to the eigenvectors of $A^TA$; based on the property that $rank(AA^T)=rank(A^TA)$. That the rank is identical implies that the number of non-zero eigenvectors is identical. Moreover, we can infer the eigenvectors of $A^TA$ from $AA^T$ and vice versa. The eigenvector decomposition of $AA^T$ is given by $AA^Tv_i = \lambda_i v_i$. In case $A$ is not a square matrix and $AA^T$ is too large to efficiently compute the eigenvectors (like it frequently occurs in covariance matrix computation), then it's easier to compute the eigenvectors of $A^TA$ given by $A^TAu_i = \lambda_i u_i$. Pre-multiplying both sides of this equation with $A$ yields
$AA^TAu_i=\lambda_iAu_i$.
Now, the originally searched eigenvectors $v_i$ of $AA^T$ can easily be obtained by $v_i:=Au_i$. Note, that the resulted eigenvectors are not yet normalized.
Special? Yes! The matrix $A^TA$ is abundant with the Least Squares Finite Element Method in Numerical Analysis:
program Demo;Output:type matrix = array of array of double;
procedure Inverteren(var b : matrix); { Matrix inversion pivots on diagonal } var pe : double; NDM,i,j,k : integer; begin NDM := Length(b); for k := 0 to NDM-1 do begin pe := b[k,k]; b[k,k] := 1; for i := 0 to NDM-1 do begin b[i,k] := b[i,k]/pe; if (i = k) then Continue; for j := 0 to NDM-1 do begin if (j = k) then Continue; b[i,j] := b[i,j] - b[i,k]*b[k,j]; end; end; for j := 0 to NDM-1 do b[k,j] := - b[k,j]/pe; b[k,k] := 1/pe; end; end;
procedure inverse(b : matrix; var q : matrix); { Matrix inversion without pivoting } var NDM,i,j,k : integer; p : matrix; h : double; begin NDM := Length(b); SetLength(p,NDM,NDM); { Transpose * original } for i := 0 to NDM-1 do begin for j := 0 to NDM-1 do begin h := 0; for k := 0 TO NDM-1 do h := h + b[k,i]b[k,j]; p[i,j] := h; end; end; Inverteren(p); SetLength(q,NDM,NDM); { (B^TB)^(-1)*B^T = B^(-1) } for i := 0 to NDM-1 do begin for j := 0 to NDM-1 do begin h := 0; for k := 0 TO NDM-1 do h := h + p[i,k]*b[j,k]; q[i,j] := h; end; end; end;
procedure test(NDM : integer); var i,j,k : integer; b,q : matrix; h : double; begin SetLength(b,NDM,NDM); Random; Random; for i := 0 to NDM-1 do begin for j := 0 to NDM-1 do begin b[i,j] := Random; end; end; inverse(b,q); for i := 0 to NDM-1 do begin for j := 0 to NDM-1 do begin h := 0; for k := 0 TO NDM-1 do h := h + q[i,k]*b[k,j]; Write(h:5:2,' '); end; Writeln; end; end;
BEGIN test(9); END.
1.00 0.00 0.00 -0.00 0.00 0.00 0.00 0.00 -0.00 0.00 1.00 0.00 -0.00 0.00 0.00 0.00 0.00 -0.00 -0.00 0.00 1.00 0.00 -0.00 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 1.00 -0.00 -0.00 -0.00 -0.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.00 -0.00 1.00 0.00 0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 1.00 -0.00 0.00 -0.00 -0.00 -0.00 0.00 -0.00 -0.00 -0.00 1.00 0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 -0.00 1.00
LATE EDIT.
Theory of Polar Decomposition is described in Wikipedia.
If attention is restricted to real-valued (non-singular square invertible) matrices, then an appropriate question and
some answers are found in Polar decomposition of real matrices.
Especially the following formula over there leaves no doubt that a matrix multiplied with its transpose IS something special:
$$
B = \left[B\left(B^TB\right)^{-1/2}\right]\left[\left(B^TB\right)^{1/2}\right]
$$
Least Squares methods (employing a matrix multiplied with its transpose) are also very useful with
Automated Balancing of Chemical Equations
One could name some properties, like if $B=AA^T$ then
$$B^T=(AA^T)^T=(A^T)^TA^T=AA^T=B,$$
so
$$\langle v,Bw\rangle=\langle Bv,w\rangle=\langle A^Tv,A^Tw\rangle.$$
Something that occurred to me while reading this answer for help with my homework is that there is a pretty common and important special case, if the linear operator A is normal, i.e. $A^TA=AA^T$. Note this is a stronger condition than saying that $A^TA$ is symmetric, which is always true. In this case, we have that $A$ is diagonalizable. As an obvious special case, $A$ is normal if $A$ is Hermitian (symmeric in the real case).
To see that $A$ normal implies $A^TA$ is diagonalizable, let $\lambda$ be a eigenvalue of $A^TA$ corresponding to the eigenvector x. Then $$(Ax,Ax)=(A^TAx,x)=\lambda(x,x)$$ and similarly $$(A^Tx,A^Tx)=(AA^Tx,x)=\lambda(x,x)$$ where I have used the normality of A. Subtracting these two equations, we have $$((A^T-A)x,(A^T-A)x)=0,$$ which by the definition of inner product implies that $$(A^T-A)x=0 =>A^Tx=Ax.$$ Note that of course this is true whenever $A=A^T$ if A is symmetric, but is a more general condition, since x is an eigenvector, rather than an arbitrary vector. Applying A to both sides of this equation, we have $$A^2x=A^TAx=\lambda x$$.
This shows that if x is an eigenvector of $A^TA$ then it is also an eigenvector of $A^2$, which in turn means it is an eigenvector of $A$ since powers of matrices share the same eigenspace. Therefore, we have constructed a full-rank set of eigenvectors of A, meaning that it is diagonalizable.
To my surprise no one mentioned yet that the root of the Gram determinant of an $n\times k$ matrix $A$ is the $k$-volume of the parallelepiped spanned by the $k$ column vectors of $A$.
The product $A^TA$ appears in a key role in the normal equations $A^TAx=A^T b$ for solving linear least squares problems.
If you have a real vector space equipped with a scalar product, and an Orthogonal matrix $A$ then $AA^T=I$ holds. A matrix is orthogonal if for the scalar product $\langle v,w \rangle = \langle Av, Aw \rangle$ holds for any $v,w \in V$
However I don't see a direct link to the Gram-Determinant.
The matrix $A$ represents a rotation if $A^TA=I$ and Det A=1 (where $A^T$ means $A$ transpose)