How to interpret a PSSM matrix from PSI-Blast - what do the numbers represent?

Question

PSI-Blast on a sequence returns a PSSM matrix file containing two matrices, one to the left, one the the right, for all amino acids:

A   R   N   D   C   Q   E   G   H   I   L   K   M   F   P   S   T   W   Y   V

What is the difference between these two matrices, how can I interpret them?

Last position-specific scoring matrix computed, weighted observed percentages rounded down, information per position, and relative weight of gapless real matches to pseudocounts
            A   R   N   D   C   Q   E   G   H   I   L   K   M   F   P   S   T   W   Y   V   A   R   N   D   C   Q   E   G   H   I   L   K   M   F   P   S   T   W   Y   V
    1 R    -2   6  -1  -2  -4   1   0  -3   0  -3  -3   2  -2  -3  -2  -1  -1  -3  -2  -3    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    2 G     0  -3   0  -2  -3  -2  -2   7  -2  -4  -4  -2  -3  -4  -2   0  -2  -3  -4  -4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    3 G     0  -3   0  -2  -3  -2  -2   7  -2  -4  -4  -2  -3  -4  -2   0  -2  -3  -4  -4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    4 G     0  -3   0  -2  -3  -2  -2   7  -2  -4  -4  -2  -3  -4  -2   0  -2  -3  -4  -4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    5 L    -2  -3  -4  -4  -1  -2  -3  -4  -3   2   4  -3   2   0  -3  -3  -1  -2  -1   1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    6 C     0  -4  -3  -4  10  -3  -4  -3  -3  -1  -1  -4  -2  -3  -3  -1  -1  -3  -3  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    7 Y    -2  -2  -2  -4  -3  -2  -2  -4   2  -2  -1  -2  -1   3  -3  -2  -2   3   8  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    8 C     0  -4  -3  -4  10  -3  -4  -3  -3  -1  -1  -4  -2  -3  -3  -1  -1  -3  -3  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
    9 R    -2   6  -1  -2  -4   1   0  -3   0  -3  -3   2  -2  -3  -2  -1  -1  -3  -2  -3    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   10 R    -2   6  -1  -2  -4   1   0  -3   0  -3  -3   2  -2  -3  -2  -1  -1  -3  -2  -3    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   11 R    -2   6  -1  -2  -4   1   0  -3   0  -3  -3   2  -2  -3  -2  -1  -1  -3  -2  -3    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   12 F    -3  -3  -3  -4  -3  -4  -4  -4  -1   0   0  -4   0   7  -4  -3  -2   1   3  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   13 C     0  -4  -3  -4  10  -3  -4  -3  -3  -1  -1  -4  -2  -3  -3  -1  -1  -3  -3  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   14 V     0  -3  -3  -4  -1  -3  -3  -4  -4   3   1  -3   1  -1  -3  -2   0  -3  -1   4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   15 C     0  -4  -3  -4  10  -3  -4  -3  -3  -1  -1  -4  -2  -3  -3  -1  -1  -3  -3  -1    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   16 V     0  -3  -3  -4  -1  -3  -3  -4  -4   3   1  -3   1  -1  -3  -2   0  -3  -1   4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   17 G     0  -3   0  -2  -3  -2  -2   7  -2  -4  -4  -2  -3  -4  -2   0  -2  -3  -4  -4    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00
   18 R    -2   6  -1  -2  -4   1   0  -3   0  -3  -3   2  -2  -3  -2  -1  -1  -3  -2  -3    0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0   0  0.00     0.00

Additionally, at the end of the file, there is additional information, what does it show exactly?

                      K         Lambda
Standard Ungapped    0.1628     0.3454
Standard Gapped      0.0410     0.2670
PSI Ungapped         0.1628     0.3454
PSI Gapped           0.0410     0.2670

Answer 1

If you read wiki carefully, you can see that PSSM is calculated in 3 steps. First frequency is calculated (how much times was the amino acid or nucleotide on that location in the motif), from that you can calculate probability (in the wiki example there were 10 sequences, so each freq. is divided by 10).

Then the log likelihood is calculated, which are the PSSM values. These are in your first matrix (rounded down). The second matrix shows how much the values are relative from your pseudocounts (I assume it was set to default = 0).

Lambda and kappa are estimated to calculate the normalized score (S') for HSP, if you never heard of this before I suggest you read the original psi-blast paper first which you can find here.