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Problem 2. (Approximate a Symmetric Matrix) 1) Let A and B be m × k matrices. Denote the entry on the ith row and jth column of A as $a_{ij}$, and the entry on the ith row and jth column of B as $b_{ij}$ ($1 \le i \le m$, $1 \le j \le k$). We can then evaluate the difference between these two matrices by calculating the sum of square differences between corresponded entries. Show that, this difference can be represented as following: $$ \sum_{i=1}^{m} \sum_{j=1}^{k} (a_{ij} - b_{ij})^2 = tr[(A - B)(A - B)'] $$ 2) Now let us focus on a $p \times p$ symmetric matrix A with spectral decomposition: $$A = \sum_{i=1}^{p} \lambda_i e_i e_i';$$ We can use a lower rank matrix B to approximate A, in particular, let us define $$B = \sum_{i=1}^{q} \lambda_i e_i e_i';$$ for $1 \le q < p$. Show that: $$tr[(A - B)(A - B)'] = \sum_{i=q+1}^{p} \lambda_i^2$$ That is, the error of approximation equals to the sum of the squares of the eigenvalues not included in B.

          Problem 2. (Approximate a Symmetric Matrix)
1) Let A and B be m × k matrices. Denote the entry on the ith row and jth column of A
as $a_{ij}$, and the entry on the ith row and jth column of B as $b_{ij}$ ($1 \le i \le m$, $1 \le j \le k$). We
can then evaluate the difference between these two matrices by calculating the sum of square
differences between corresponded entries. Show that, this difference can be represented as
following:
$$
\sum_{i=1}^{m} \sum_{j=1}^{k} (a_{ij} - b_{ij})^2 = tr[(A - B)(A - B)']
$$
2) Now let us focus on a $p \times p$ symmetric matrix A with spectral decomposition:
$$A = \sum_{i=1}^{p} \lambda_i e_i e_i';$$
We can use a lower rank matrix B to approximate A, in particular, let us define
$$B = \sum_{i=1}^{q} \lambda_i e_i e_i';$$
for $1 \le q < p$. Show that:
$$tr[(A - B)(A - B)'] = \sum_{i=q+1}^{p} \lambda_i^2$$
That is, the error of approximation equals to the sum of the squares of the eigenvalues
not included in B.

Problem 2. (Approximate a Symmetric Matrix)
1) Let A and B be m × k matrices. Denote the entry on the ith row and jth column of A
as aij, and the entry on the ith row and jth column of B as bij (1 ≤ i ≤ m, 1 ≤ j ≤ k). We
can then evaluate the difference between these two matrices by calculating the sum of square
differences between corresponded entries. Show that, this difference can be represented as
following:

∑i=1^m∑j=1^k (aij - bij)^2 = tr[(A - B)(A - B)']

2) Now let us focus on a p × p symmetric matrix A with spectral decomposition:

A = ∑i=1^p ei ei';

We can use a lower rank matrix B to approximate A, in particular, let us define

B = ∑i=1^q ei ei';

for 1 ≤ q < p. Show that:

tr[(A - B)(A - B)'] = ∑i=q+1^p^2

That is, the error of approximation equals to the sum of the squares of the eigenvalues
not included in B.

Added by Ashlee K.

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